A few example designs and data sets for this module are available in the R package apts.doe, which can be installed from GitHub

library(devtools)
install_github("statsdavew/apts.doe", quiet = T)
library(apts.doe)

References will be provided throughout but some good general purpose texts are

• Santner, Williams and Notz (2003). The Design and Analysis of Computer Experiments. Springer.
• Atkinson, Donev and Tobias (2007). Optimum Experimental Design, with SAS. OUP
• Wu and Hamada (2009). Experiments: Planning, Analysis, and Parameter Design Optimization (2nd ed.). Wiley.
• Morris (2011). Design of Experiments: An Introduction based on Linear Models. Chapman and Hall/CRC Press.

These notes and other resources can be found at https://statsdavew.github.io/apts.doe/

• Observational studies
• Sample surveys
• Designed experiments

Why do we experiment?

• key to the scientific method
  (hypothesis – experiment – observe – infer – conclude)

• potential to establish causality …

• … and to understand/improve complex systems depending on many factors

• comparison of treatments, factor screening, prediction, optimisation, …

Design of experiments: a statistical approach to the arrangement of the operational details of the experiment (eg sample size, specific experimental conditions investigated, …) so that the quality of the answers to be derived from the data is as high as possible.

Consider an experiment to compare two treatments (eg drugs, diets, fertilisers).

We have \(n\) subjects (eg people, mice, plots of land), each of which can be assigned to one of the two treatments.

A response (eg protein measurement, weight, yield) is then measured from each subject.

Question: How should the two treatments be assigned to the subjects to gain the most precise inference about the difference in expected response from the two treatments.

Assume a linear model for the response \[ y_i = \beta_0 + \beta_1 x_i + \varepsilon_i\,,\qquad i=1,\ldots,n\,, \] with \(\varepsilon_i\sim N(0, \sigma^2)\) independently, \(\beta_0,\beta_1\) unknown parameters and \[ x_i = \left\{ \begin{array}{cc} -1 & \mbox{if treatment 1 is applied to subject $i$}\,, \\ +1 & \mbox{if treatment 2 is applied to subject $i$} \end{array} \right. \] The difference in expected response between treatment 1 and 2 is \[ E(y_i\,|\, x_i = +1) - E(y_i\,|\, x_i = -1) = \beta_0 + \beta_1 - \beta_0 + \beta_1 = 2\beta_1 \] So we need the most precise possible estimator of \(\beta_1\)

Both \(\beta_0\) and \(\beta_1\) can be estimated using least squares (or equivalently maximum likelihood).

Writing \[ {\boldsymbol{y}}= X{\boldsymbol{\beta}}+ {\boldsymbol{\varepsilon}}\,, \] we obtain estimators \[ \hat{{\boldsymbol{\beta}}} = \left(X^{\mathrm{T}}X\right)^{-1}X^{\mathrm{T}}{\boldsymbol{y}}\] with \[ \mbox{Var}(\hat{{\boldsymbol{\beta}}}) = \left(X^{\mathrm{T}}X\right)^{-1}\sigma^2 \] In this simple example, we are interesting in estimating \(\beta_1\), and we have \[ \begin{split} \mbox{Var}(\hat{\beta_1}) & = \frac{n\sigma^2}{n\sum x_i^2 - \left(\sum x_i\right)^2}\\ & = \frac{n\sigma^2}{n^2 - \left(\sum x_i\right)^2} \end{split} \]

Hence, we need to pick \(x_1,\ldots,x_n\) to minimise \(\left(\sum x_i\right)^2 = (n_1 - n_2)^2\)

• denote as \(n_1\) the number of subjects assigned to treatment 1, and \(n_2\) the number assigned to treatment 2, with \(n_1+n_2 = n\)
• it is obvious that \(\sum x_i = 0\) if and only if \(n_1 = n_2\)

Assuming \(n\) is even, the "optimal design" has \(n_1 = n_2 = n/2\)

For \(n\) odd, let \(n_1 = \frac{n+1}{2}\) and \(n_2 = \frac{n-1}{2}\)

We can assess a designs, labelled \(\xi\), via its efficiency relative to the optimal design \(\xi^\star\): \[ \mbox{Eff($\xi$)} = \frac{\mbox{Var}(\hat{\beta_1}\,|\,\xi^\star)}{\mbox{Var}(\hat{\beta_1}\,|\,\xi)} \]

n <- 50
eff <- function(n1) 1 - ((2 * n1 - n) / n)^2
curve(eff, from = 0, to = n, ylab = "Eff", xlab = expression(n[1]))

• Treatment – entities of scientific interest to be studied in the experiment
  eg varieties of crop, doses of a drug, combinations of temperature and pressure

• Unit – smallest subdivision of the experimental material such that two units may receive different treatments
  eg plots of land, subjects in a clinical trial, samples of reagent

• Run – application of a treatment to a unit

Fabrication of integrated circuits (see Wu and Hamada 2009)

• an initial step in fabricating integrated circuits is the growth of an epitaxial layer on polished silicon wafers via chemical deposition

Unit

• set of six wafers (mounted in a rotating cylinder)

Treatment

• combination of settings of the factors
  • A : rotation method (\(x_1\))
  • B : nozzle position (\(x_2\))
  • C : deposition temperature (\(x_3\))
  • D : deposition time (\(x_4\))

\[ y_{ij} = \tau_i + \varepsilon_{ij}\,,\qquad i=1,\ldots,t;\,j=1,\ldots,n_i\,, \] where

• \(y_{ij}\) : measured response from the \(j\)th unit to which treatment \(i\) has been applied

• \(\tau_i\) : treatment effect (expected response from application of the \(i\)th treatment)

• \(\varepsilon_{ij}\) : random deviation from the expected response [typically \(\sim N(0,\sigma^2)\)]

The aims of the experiment are achieved by estimating comparisons between the treatment effects, \(\tau_k - \tau_l\).

Experimental precision and accuracy are largely obtained through control and comparison.

Three key model assumptions are:

• additivity (response = treatment effect + unit effect)
• constancy of treatment effects (treatment effect does not depend on the unit to which it is applied)
• no interference between units (the effect of a treatment applied to unit \(j\) does not depend on the treatment applied to any other unit)

See Dasgupta, Pillai, and Rubin (2015) for discussion of these assumptions for factorial experiments

Stratification (blocking)

• account for systematic differences between batches of experimental units by arranging them in homogeneous sets (blocks)
  • if the same treatment was applied to all units, within-block variation in the response would be much less than between-block
  • compare treatments within the same block and hence eliminate block effects

Replication

• the application of each treatment to multiple experimental units
  • provides an estimate of experimental error against which to judge treatment differences
  • reduces the variance of the estimators of treatment differences

Randomisation

• we randomise features such as the allocation of units to treatments, the order in which treatments are applied, …
  • protects against lurking (uncontrolled) variables (model-robust) and subjectively in the allocation of treatments to units

Randomisation is perhaps the key principle in the design of experiments

• it protects against model misspecification (bias), and hence allows causality to be established
  • a clear difference between treatments can only be an accident of the randomisation or a consequence of the treatments
• unbiased estimation of \(\tau\) and \(\sigma^2\), even if the errors are not normally distributed
• exact tests for differences between treatment effects are available (Basu 1980)

Without randomisation, unobserved confounders (\(U\)) can induce a dependency between
the response (\(Y\)) and treatment (\(T\)) cf Cox and Reid (2000), p.35

With randomisation, unobserved confounders (\(U\)) are independent of the treatment (\(T\)). Marginalisation over \(U\) does not induce an edge between \(T\) and \(Y\) cf Cox and Reid (2000), p.35

Fabrication of integrated circuits (Wu and Hamada 2009, p155)

Treatment

• combination of settings of the factors
  • A : rotation method (\(x_1\))
  • B : nozzle position (\(x_2\))
  • C : deposition temperature (\(x_3\))
  • D : deposition time (\(x_4\))

Assume each factor has two-levels, coded -1 and +1

Each factor has two levels \(x_k = \pm 1,\, k=1,\ldots,4\)

A treatment is then defined as a combination of four values of \(-1, +1\)

• eg \(x_1 = -1, x_2 = -1, x_3 = +1, x_4 = -1\)
• specifies a setting of the process

Assume each treatment effect is determined by a regression model in the four factors, eg \[ \tau({\boldsymbol{x}}) = \beta_0 + \sum_{i=1}^4\beta_ix_i + \sum_{j=1}^4\sum_{i>j}^4\beta_{ij}x_ix_j \]

with(cirfab, cirfab[order(x1, x2, x3, x4), ])

##    x1 x2 x3 x4     ybar
## 2  -1 -1 -1 -1 13.58983
## 1  -1 -1 -1  1 14.59000
## 4  -1 -1  1 -1 14.04983
## 3  -1 -1  1  1 14.24000
## 6  -1  1 -1 -1 13.94000
## 5  -1  1 -1  1 14.65000
## 8  -1  1  1 -1 14.14017
## 7  -1  1  1  1 14.40000
## 10  1 -1 -1 -1 13.72000
## 9   1 -1 -1  1 14.67000
## 12  1 -1  1 -1 13.90000
## 11  1 -1  1  1 13.84017
## 14  1  1 -1 -1 13.87983
## 13  1  1 -1  1 14.56000
## 16  1  1  1 -1 14.11017
## 15  1  1  1  1 14.30000

• treatments in standard order

• \(\bar{y}\) - average response from the six wafers

\[ \boldsymbol{Y} = X\boldsymbol{\beta} + \boldsymbol{\varepsilon}\,,\qquad \boldsymbol{\varepsilon}\sim N(\boldsymbol{0}, \sigma^2I)\,,\qquad \hat{\boldsymbol{\beta}} = \left(X^\mathrm{T}X\right)^{-1}X^\mathrm{T}\boldsymbol{Y} \]

• model matrix \(X\) has columns corresponding to intercept, linear and cross-product terms

• information matrix \(X^\mathrm{T}X = nI\)

• regression coefficients are estimated by independent contrasts in the data

cirfab.lm <- lm(ybar ~ (.) ^ 2, data = cirfab)
coef(cirfab.lm)

##  (Intercept)           x1           x2           x3           x4        x1:x2 
## 14.161250000 -0.038729167  0.086270833 -0.038708333  0.245020833  0.003708333 
##        x1:x3        x1:x4        x2:x3        x2:x4        x3:x4 
## -0.046229167 -0.025000000  0.028770833 -0.015041667 -0.172520833

Main effect of \(x_k\): \[ [\text{Avg. response when $x_k = 1$}]\, -\, [\text{Avg. response when $x_k = -1$}] \]

Interaction between \(x_j\) and \(x_k\): \[ [\text{Avg. response when $x_jx_k= 1$}]\, -\, [\text{Avg. response when $x_jx_k = -1$}] \]

Higher-order interactions defined similarly

Assuming -1,+1 coding, there is a straightforward relationship between factorial effects and regression coefficients

• main effect of \(x_k\) is equal to \(2\beta_k\)
• interaction between \(x_j\) and \(x_k\) is equal to \(2\beta_{jk}\)

Using the effects package:

library(effects)
plot(Effect("x1", cirfab.lm), main = "", rug = F, ylim = c(13.5, 14.5), aspect = 1)
plot(Effect("x2", cirfab.lm), main = "", rug = F, ylim = c(13.5, 14.5), aspect = 1)
plot(Effect("x3", cirfab.lm), main = "", rug = F, ylim = c(13.5, 14.5), aspect = 1)
plot(Effect("x4", cirfab.lm), main = "", rug = F, ylim = c(13.5, 14.5), aspect = 1)

plot(Effect(c("x3", "x4"), cirfab.lm), main = "", rug = F, ylim = c(13.5, 15), 
     x.var = "x4")

\(X^\mathrm{T}X = nI \Rightarrow \hat{\boldsymbol{\beta}}\) are independently normally distributed with equal variance

Hence, we can treat the identification of important effects (ie large \(\beta\)) as an outlier identification problem

• plot (absolute) ordered factorial effects against (absolute) quantiles from a standard normal
• outlying effects are identified as important

Cuthbert (1959)

Using the FrF2 package

library(FrF2)
par(pty = "s", mar = c(8, 4, 1, 2))
DanielPlot(cirfab.lm, main = "", datax = F, half = T)

An unreplicated factorial design provides no model-independent estimate of \(\sigma^2\) (Gilmour and Trinca 2012)

• any unsaturated model does provide an estimate, but it may be biased by ignored (significant) model terms
• this is one reason why graphical (or associated) analysis methods are popular

Replication also increases the power of the design

• common to replicate a centre point
• allows a portmanteau test of curvature
• allows unbiased estimation of \(\sigma^2\)

Effect sparsity

• the number of important effects in a factorial experiment is small relative to the total number of effects investigated (cf Box and Meyer 1986)

Effect hierarchy

• lower-order effects are more likely to be important than higher-order effects
• effects of the same order are equally likely to be important

Effect heredity

• interactions where at least one parent main effect is important are more likely to be important themselves

Wu and Hamada (2009), pp.172–172

Factorial designs can require a large number of runs for only a moderate number of factors (\(2^5 = 32\))

Resource constraints (eg cost) may mean not all \(2^m\) combinations can be run

Lots of degrees of freedom are devoted to estimating higher-order interactions

• eg in a \(2^5\) experiment, 16 degrees of freedom are used to estimate three-factor and higher-order interactions
• principles of effect hierarchy and sparsity suggest may be wasteful

Need to trade-off what you want to estimate against the number of runs you can afford

Production of bacteriocin (Morris 2011, p231)

• bacteriocin is a natural food preservative frown from bacteria

Unit

• a single bio-reaction

Treatment: combination of settings of the factors

• A: amount of glucose (\(x_1\))
• B: initial inoculum size (\(x_2\))
• C: level of aeration (\(x_3\))
• D: temperature (\(x_4\))
• E: amount of sodium (\(x_5\))

Assume each factor has two-levels, coded -1 and +1

Find an \(n=8\) run design using FrF2

bact.design <- FrF2(8, 5, factor.names = paste0("x", 1:5), 
     generators = list(c(1, 3), c(2, 3)), randomize = F, alias.info = 3)
bact.design

##   x1 x2 x3 x4 x5
## 1 -1 -1 -1  1  1
## 2  1 -1 -1 -1  1
## 3 -1  1 -1  1 -1
## 4  1  1 -1 -1 -1
## 5 -1 -1  1 -1 -1
## 6  1 -1  1  1 -1
## 7 -1  1  1 -1  1
## 8  1  1  1  1  1
## class=design, type= FrF2.generators

• \(8\) = \(32/4\) = \(2^5/2^2\) = \(2^{5-2}\)
• we need a principled way of choosing one-quarter of the runs from the factorial design that leads to clarity in the analysis

Assuming the number of runs is a power of two, \(n = 2^{k-q}\), we can construct \(2^{k-q} -1\) orthogonal vectors (with inner product zero), spanned by \(k-q = \log_2(n)\) vectors

• construct the full factorial design for \(k-q\) factors
• assign the remaining \(q\) factors to interaction columns

model.matrix(~ (x1 + x2 + x3) ^ 3, bact.design[, 1:3])[, ]

##   (Intercept) x11 x21 x31 x11:x21 x11:x31 x21:x31 x11:x21:x31
## 1           1  -1  -1  -1       1       1       1          -1
## 2           1   1  -1  -1      -1      -1       1           1
## 3           1  -1   1  -1      -1       1      -1           1
## 4           1   1   1  -1       1      -1      -1          -1
## 5           1  -1  -1   1       1      -1      -1           1
## 6           1   1  -1   1      -1       1      -1          -1
## 7           1  -1   1   1      -1      -1       1          -1
## 8           1   1   1   1       1       1       1           1

The design has been deliberately chosen so that

• \(x_4 = x_1x_3\)
• \(x_5 = x_2x_3\)

[\(x_1x_2\) is shorthand for the Hadamard (Schur or entry wise) product of two vectors, \(x_1\circ x_2\)]

What other consequences are there?

• \(x_4x_5 = x_1x_3x_2x_3 = x_1x_2x_3^2\)
• the product of any column with itself is the constant column (the identity)
• hence, \(x_4x_5 = x_1x_2\)

Now we can obtain the defining relation \(\ldots\)

• \(I = x_1x_3x_4 = x_2x_3x_5 = x_1x_2x_4x_5\)

\(\ldots\) and the complete aliasing scheme

• \(x_1 = x_3x_4 = x_1x_2x_3x_5 = x_2x_4x_5\)
• \(x_2 = x_1x_2x_3x_4 = x_3x_5 = x_1x_4x_5\)
• \(x_3 = x_1x_4 = x_2x_5 = x_1x_2x_3x_4x_5\)
• \(x_4 = x_1x_3 = x_2x_3x_4x_5 = x_1x_2x_5\)
• \(x_5 = x_1x_3x_4x_5 = x_2x_3 = x_1x_2x_4\)
• \(x_1x_2 = x_2x_3x_4 = x_1x_3x_5 = x_4x_5\)
• \(x_1x_5 = x_3x_4x_5 = x_1x_2x_3 = x_2x_4\)

FrF2 will summarise the aliasing amongst main effects and two- and three-factor interactions.

design.info(bact.design)$aliased 

## $legend
## [1] "A=x1" "B=x2" "C=x3" "D=x4" "E=x5"
## 
## $main
## [1] "A=CD=BDE" "B=CE=ADE" "C=AD=BE"  "D=AC=ABE" "E=BC=ABD"
## 
## $fi2
## [1] "AB=DE=ACE=BCD" "AE=BD=ABC=CDE"
## 
## $fi3
## [1] "ACD=BCE"

What is the consequence of this aliasing?

If more than one effect in each alias string is non-zero, the least squares estimators will be biased

• assumed model \({\boldsymbol{Y}}= X_1{\boldsymbol{\beta}}_1 + {\boldsymbol{\varepsilon}}\)
• true model \({\boldsymbol{Y}}= X_1{\boldsymbol{\beta}}_1 + X_2{\boldsymbol{\beta}}_2 + {\boldsymbol{\varepsilon}}\)

\[ \begin{split} E\left(\hat{{\boldsymbol{\beta}}}_1\right) & = \left(X_1^\mathrm{T}X_1\right)^{-1}X^\mathrm{T}_1E({\boldsymbol{Y}}) \\ & = \left(X^{\mathrm{T}}_1X_1\right)^{-1}X_1^{\mathrm{T}}\left(X_1{\boldsymbol{\beta}}_1 + X_2{\boldsymbol{\beta}}_2\right) \\ & = \beta_1 + \left(X_1^{\mathrm{T}}X_1\right)^{-1}X_1^{\mathrm{T}}X_2{\boldsymbol{\beta}}_2 \\ & = {\boldsymbol{\beta}}_1 + A{\boldsymbol{\beta}}_2\\ \end{split} \]

\(A\) is the alias matrix

• if the columns of \(X_1\) and \(X_2\) are not orthogonal, \(\hat{{\boldsymbol{\beta}}}_1\) is biased

For the \(2^{5-2}\) example:

• \(X_1\) is an \(8\times 6\) matrix with columns for the intercept and five linear terms ("main effects")
• \(X_2\) is an \(8\times 10\) matrix with columns for the 10 product terms ("two-factor interactions")

\[ A = \left( \begin{array}{cccccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \end{array} \right) \] For a regular design, the matrix \(A\) will only have entries 0, \(\pm 1\) (no aliasing or complete aliasing)

The transpose of the alias matrix is provided by the alias function.

ff.alias <- alias(y ~ (.)^2, data = data.frame(bact.design, y = vector(length = 8)))
ff.alias$Complete

##         (Intercept) x11 x21 x31 x41 x51 x11:x21 x11:x51
## x21:x31 0           0   0   0   0   1   0       0      
## x21:x41 0           0   0   0   0   0   0       1      
## x21:x51 0           0   0   1   0   0   0       0      
## x31:x41 0           1   0   0   0   0   0       0      
## x31:x51 0           0   1   0   0   0   0       0      
## x41:x51 0           0   0   0   0   0   1       0      
## x11:x31 0           0   0   0   1   0   0       0      
## x11:x41 0           0   0   1   0   0   0       0

Typically, in a first experiment, fractional factorial designs are used in screening

• investigate which of many factors have a substantive effect on the response
• main effects and two-factor interactions
• centre points to check for curvature

At second and later stages, augment the design

• to resolve ambiguities due to the aliasing of factorial effects ("break the alias strings")
• to allow estimation of curvature and prediction from a more complex model

Regular fractional factorial designs have the number of runs equal to a power of the number of levels

• eg \(2^{5-2}\), \(3^{3-1}\times 2\)
• this inflexibility in run sizes can be a problem in practical experiments

Non-regular designs can have any number of runs (usually with \(n>p\), the number of parameters to be estimated)

Often the clarity provided by a regular design is lost

• no defining relation or straightforward aliasing scheme
• partial aliasing and fractional entries in \(A\)

One approach to finding non-regular designs is via a design optimality criterion

Notation: let \(\xi = [{\boldsymbol{x}}_1,\ldots,{\boldsymbol{x}}_n]\) denote a design (choice of treatments and their replications)

Assuming the model \({\boldsymbol{Y}}= X{\boldsymbol{\beta}}+ {\boldsymbol{\varepsilon}}\), with \({\boldsymbol{\varepsilon}}\sim N(0, \sigma^2I_n)\), a \(D\)-optimal design maximises \[ \phi(\xi) = \mathrm{det}\left(X^{\mathrm{T}}X\right) \]

That is, a \(D\)-optimal design maximises the determinant of the (expected) Fisher information matrix

• equivalent to minimising the volume of the joint confidence ellipsoid for \({\boldsymbol{\beta}}\)

Also useful to define a Bayesian version, with \(R\) a prior precision matrix \[ \phi_B(\xi) = \mathrm{det}\left(X^{\mathrm{T}}X + R\right) \] (See later)

\(D\)-optimal designs are model dependent

• if the model (ie the columns of \(X\)) changes, the optimal design may change
• model-robust design is an active area of research

\(D\)-optimality promotes orthogonality in the \(X\) matrix

• if there are sufficient runs, the \(D\)-optimal design will usually be orthogonal
• for particular models and choices of \(n\), regular fractional factorial designs are \(D\)-optimal

There are many other optimality criteria, tailored to other experimental goals

• prediction, model discrimination, space-filling, …

\(k=11\) factors in \(n=12\) runs, first-order (main effects) model (Plackett and Burman 1946)

A particular \(D\)-optimal design is the following orthogonal array

Using the pb function in the FrF2 package:

pb.design <- pb(12, factor.names = paste0("x", 1:11))
pb.design

##    x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11
## 1  -1  1 -1  1  1 -1  1  1  1  -1  -1
## 2  -1 -1  1 -1  1  1 -1  1  1   1  -1
## 3  -1 -1 -1  1 -1  1  1 -1  1   1   1
## 4   1  1  1 -1 -1 -1  1 -1  1   1  -1
## 5  -1  1  1 -1  1  1  1 -1 -1  -1   1
## 6   1 -1 -1 -1  1 -1  1  1 -1   1   1
## 7  -1  1  1  1 -1 -1 -1  1 -1   1   1
## 8  -1 -1 -1 -1 -1 -1 -1 -1 -1  -1  -1
## 9   1 -1  1  1 -1  1  1  1 -1  -1  -1
## 10  1  1 -1  1  1  1 -1 -1 -1   1  -1
## 11  1 -1  1  1  1 -1 -1 -1  1  -1   1
## 12  1  1 -1 -1 -1  1 -1  1  1  -1   1
## class=design, type= pb

This 12-run PB design is probably the most studied non-regular design

• orthogonal columns
• complex aliasing between main effects and two-factor interactions

pb.alias <- alias(y ~ (.)^2, data = data.frame(pb.design, y = vector(length = 12)))
head(pb.alias$Complete, n = 15)

##          (Intercept) x11  x21  x31  x41  x51  x61  x71  x81  x91  x101 x111
## x11:x21     0           0    0 -1/3 -1/3 -1/3  1/3 -1/3 -1/3  1/3  1/3 -1/3
## x11:x31     0           0 -1/3    0  1/3 -1/3 -1/3  1/3 -1/3  1/3 -1/3 -1/3
## x11:x41     0           0 -1/3  1/3    0  1/3  1/3 -1/3 -1/3 -1/3 -1/3 -1/3
## x11:x51     0           0 -1/3 -1/3  1/3    0 -1/3 -1/3 -1/3 -1/3  1/3  1/3
## x11:x61     0           0  1/3 -1/3  1/3 -1/3    0 -1/3  1/3 -1/3 -1/3 -1/3
## x11:x71     0           0 -1/3  1/3 -1/3 -1/3 -1/3    0  1/3 -1/3  1/3 -1/3
## x11:x81     0           0 -1/3 -1/3 -1/3 -1/3  1/3  1/3    0 -1/3 -1/3  1/3
## x11:x91     0           0  1/3  1/3 -1/3 -1/3 -1/3 -1/3 -1/3    0 -1/3  1/3
## x11:x101    0           0  1/3 -1/3 -1/3  1/3 -1/3  1/3 -1/3 -1/3    0 -1/3
## x11:x111    0           0 -1/3 -1/3 -1/3  1/3 -1/3 -1/3  1/3  1/3 -1/3    0
## x21:x31     0        -1/3    0    0 -1/3 -1/3 -1/3  1/3 -1/3 -1/3  1/3  1/3
## x21:x41     0        -1/3    0 -1/3    0  1/3 -1/3 -1/3  1/3 -1/3  1/3 -1/3
## x21:x51     0        -1/3    0 -1/3  1/3    0  1/3  1/3 -1/3 -1/3 -1/3 -1/3
## x21:x61     0         1/3    0 -1/3 -1/3  1/3    0 -1/3 -1/3 -1/3 -1/3  1/3
## x21:x71     0        -1/3    0  1/3 -1/3  1/3 -1/3    0 -1/3  1/3 -1/3 -1/3

Screening designs with fewer runs than factors (see Woods and Lewis 2017)

• can't use ordinary least squares/maximum likelihood as \(X\) does not have full column rank
• Bayesian \(D\)-optimality with \(R = [0\,|\, \tau I_m]\)

Supersaturated experiment used by GlaxoSmithKline in the development of a new oncology drug

• \(k=16\) factors: e.g. temperature, solvent amount, reaction time
• \(n=10\) runs
• Bayesian \(D\)-optimal design with \(\tau = 0.2\)

ssd

##    x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16
## 1   1  1 -1  1  1 -1 -1 -1 -1  -1  -1   1   1   1   1   1
## 2   1  1  1 -1 -1 -1 -1 -1  1   1  -1  -1   1  -1  -1   1
## 3  -1 -1  1 -1  1 -1  1 -1 -1   1  -1  -1  -1   1   1  -1
## 4  -1  1  1  1  1 -1 -1  1 -1  -1   1  -1  -1  -1  -1  -1
## 5  -1 -1 -1 -1 -1  1 -1  1  1  -1  -1  -1  -1  -1   1   1
## 6   1  1  1 -1  1  1 -1  1  1   1   1   1   1   1   1  -1
## 7  -1 -1  1  1 -1 -1  1 -1  1  -1   1   1   1  -1   1  -1
## 8   1 -1 -1  1  1  1  1  1 -1   1  -1   1   1  -1  -1  -1
## 9  -1  1 -1  1 -1  1  1 -1  1   1   1  -1  -1   1  -1   1
## 10  1 -1  1 -1 -1 -1  1  1 -1  -1   1   1  -1   1  -1   1

Partial aliasing between main effects

Heatmap of column correlations:

library(fields)
par(mar=c(8,2,0,0))
image.plot(1:16,1:16, cor(ssd), zlim = c(-1, 1), xlab = "Factors", 
           ylab = "", asp = 1, axes = F)
axis(1, at = seq(2, 16, by = 2), line = .5)
axis(2, at = seq(2, 16, by = 2), line = -5)

Analysis via regularised (shrinkage) methods (eg lasso, Dantzig selector; see APTS High Dimensional Statistics)

• small coefficients shrunk to zero

Now consider a more general class of models (cf preliminary material).

Let \({\boldsymbol{y}}= (y_1,\ldots,y_n)^{\mathrm{T}}\) be iid observations from a distribution with density/mass function \(\pi(y_i\,;\,{\boldsymbol{\theta}},{\boldsymbol{x}}_i)\)

• \({\boldsymbol{\theta}}\) is a \(q-\)vector of unknown parameters
• \({\boldsymbol{x}}_i =(x_{1i},\ldots,x_{ki})^{\mathrm{T}}\) is a vector of values of \(k\) controllable variables.

The (expected) information matrix \[ M({\boldsymbol{\theta}}) = E_y\left[-\frac{\partial^2l({\boldsymbol{\theta}})}{\partial{\boldsymbol{\theta}}\partial{\boldsymbol{\theta}}^{\mathrm{T}}}\right] \] is an important quantity for design, where \(l({\boldsymbol{\theta}}) = \sum_{i=1}^n\log\pi(y_i;\,{\boldsymbol{\theta}},{\boldsymbol{x}}_i)\) (the log-likelihood).

• \(M({\boldsymbol{\theta}})\) is the (asymptotic) precision for the maximum likelihood estimators \(\hat{{\boldsymbol{\theta}}}\).
• \(M({\boldsymbol{\theta}})\) is also an asymptotic approximation to the posterior precision for \({\boldsymbol{\theta}}\) in a Bayesian analysis.

Example 1: Compartmental model \[ y_i \sim N\left(c({\boldsymbol{\theta}})\mu({\boldsymbol{\theta}};\,x_i), \sigma^2\nu({\boldsymbol{\theta}};\,x_i)\right)\,,\quad x_i\in[0,24]\,, \] with \[ \mu({\boldsymbol{\theta}};\,x) = \exp(-\theta_1x)-exp(-\theta_2x)\,,\quad c({\boldsymbol{\theta}}) = \frac{400\theta_2}{\theta_3(\theta_2-\theta_1)}\,,\quad \nu({\boldsymbol{\theta}};\,x) = 1 + \frac{\tau^2}{\sigma^2}c({\boldsymbol{\theta}})^2\mu({\boldsymbol{\theta}};\,x)\,, \] for \(\theta_1, \theta_2, \theta_3, \tau^2, \sigma^2>0\).

Prior distributions (for later use):

• \(\log\theta_i\sim N(m_i, 0.05)\), with \(m_1 = \log 0.1, m_2 = 0, m_3 = \log 20\)

Ryan et al. (2014)

comp <- function(x, theta, D = 400) {
    mu <- exp(-theta[1] * x) - exp(-theta[2] * x)
    c <- (D / theta[3]) * (theta[2]) / (theta[2] - theta[1])
    c * mu }
theta <- c(.1, 1, 20)
M <- 100
par(mar = c(6, 4, 0, 1) + .1)
lapply(1:M, function(l) {
  thetat <- rlnorm(3,log(theta),rep(0.05,3))
  curve(comp(x, theta = thetat), from = 0, to = 24, ylab = "Expected concentration", 
        xlab = "Time", ylim = c(0, 20), xlim = c(0, 24), add = l!=1) })

## [[1]]
## [[1]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[1]]$y
##   [1]  0.000000  4.229282  7.423523  9.813383 11.578615 12.859288 13.764536
##   [8] 14.379380 14.770043 14.988097 15.073694 15.058081 14.965570 14.815064
##  [15] 14.621252 14.395538 14.146766 13.881790 13.605906 13.323204 13.036828
##  [22] 12.749193 12.462139 12.177066 11.895024 11.616799 11.342968 11.073944
##  [29] 10.810018 10.551382 10.298156 10.050400  9.808131  9.571334  9.339967
##  [36]  9.113968  8.893263  8.677767  8.467387  8.262024  8.061577  7.865944
##  [43]  7.675018  7.488697  7.306875  7.129449  6.956316  6.787377  6.622532
##  [50]  6.461683  6.304736  6.151597  6.002174  5.856378  5.714122  5.575319
##  [57]  5.439888  5.307745  5.178811  5.053008  4.930262  4.810496  4.693640
##  [64]  4.579622  4.468374  4.359828  4.253918  4.150582  4.049755  3.951378
##  [71]  3.855391  3.761735  3.670354  3.581193  3.494198  3.409317  3.326497
##  [78]  3.245689  3.166844  3.089915  3.014854  2.941617  2.870158  2.800436
##  [85]  2.732407  2.666031  2.601268  2.538077  2.476422  2.416264  2.357568
##  [92]  2.300297  2.244418  2.189896  2.136699  2.084794  2.034150  1.984736
##  [99]  1.936522  1.889480  1.843580
## 
## 
## [[2]]
## [[2]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[2]]$y
##   [1]  0.000000  4.418395  7.725933 10.178365 11.973058 13.262314 14.163646
##   [8] 14.767718 15.144464 15.347818 15.419368 15.391168 15.287916 15.128630
##  [15] 14.927950 14.697130 14.444818 14.177646 13.900694 13.617846 13.332057
##  [22] 13.045576 12.760097 12.476896 12.196921 11.920868 11.649244 11.382406
##  [29] 11.120597 10.863974 10.612628 10.366600 10.125892  9.890476  9.660305
##  [36]  9.435314  9.215427  9.000559  8.790619  8.585514  8.385146  8.189417
##  [43]  7.998228  7.811480  7.629075  7.450916  7.276907  7.106955  6.940965
##  [50]  6.778848  6.620513  6.465874  6.314845  6.167341  6.023282  5.882587
##  [57]  5.745178  5.610977  5.479911  5.351907  5.226892  5.104797  4.985553
##  [64]  4.869095  4.755358  4.644277  4.535791  4.429839  4.326361  4.225301
##  [71]  4.126602  4.030208  3.936066  3.844122  3.754327  3.666629  3.580980
##  [78]  3.497331  3.415636  3.335850  3.257927  3.181825  3.107500  3.034911
##  [85]  2.964018  2.894781  2.827162  2.761121  2.696624  2.633633  2.572114
##  [92]  2.512031  2.453352  2.396044  2.340074  2.285412  2.232027  2.179889
##  [99]  2.128968  2.079237  2.030668
## 
## 
## [[3]]
## [[3]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[3]]$y
##   [1]  0.000000  3.611449  6.441638  8.643518 10.340429 11.631819 12.597869
##   [8] 13.303219 13.799970 14.130111 14.327477 14.419323 14.427599 14.369974
##  [15] 14.260668 14.111118 13.930515 13.726242 13.504222 13.269203 13.024984
##  [22] 12.774601 12.520471 12.264520 12.008271 11.752927 11.499430 11.248517
##  [29] 11.000757 10.756581 10.516317 10.280204 10.048413  9.821057  9.598209
##  [36]  9.379903  9.166149  8.956931  8.752220  8.551970  8.356126  8.164626
##  [43]  7.977400  7.794376  7.615477  7.440624  7.269738  7.102737  6.939541
##  [50]  6.780070  6.624242  6.471980  6.323204  6.177838  6.035805  5.897030
##  [57]  5.761440  5.628963  5.499529  5.373068  5.249513  5.128797  5.010855
##  [64]  4.895624  4.783043  4.673049  4.565584  4.460590  4.358010  4.257789
##  [71]  4.159872  4.064207  3.970741  3.879425  3.790209  3.703045  3.617885
##  [78]  3.534683  3.453395  3.373976  3.296383  3.220575  3.146510  3.074149
##  [85]  3.003451  2.934380  2.866897  2.800966  2.736551  2.673617  2.612131
##  [92]  2.552059  2.493368  2.436027  2.380005  2.325271  2.271796  2.219550
##  [99]  2.168506  2.118636  2.069913
## 
## 
## [[4]]
## [[4]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[4]]$y
##   [1]  0.000000  4.017578  7.083469  9.402838 11.137056 12.413032 13.330565
##   [8] 13.968127 14.387408 14.636898 14.754703 14.770756 14.708567 14.586590
##  [15] 14.419299 14.218046 13.991718 13.747271 13.490139 13.224560 12.953831
##  [22] 12.680514 12.406587 12.133572 11.862637 11.594666 11.330323 11.070102
##  [29] 10.814359 10.563347 10.317234 10.076128  9.840083  9.609118  9.383219
##  [36]  9.162352  8.946465  8.735492  8.529358  8.327982  8.131276  7.939151
##  [43]  7.751513  7.568268  7.389324  7.214585  7.043958  6.877351  6.714672
##  [50]  6.555832  6.400742  6.249314  6.101464  5.957109  5.816165  5.678555
##  [57]  5.544198  5.413019  5.284942  5.159895  5.037806  4.918605  4.802224
##  [64]  4.688597  4.577658  4.469343  4.363592  4.260342  4.159536  4.061114
##  [71]  3.965022  3.871203  3.779604  3.690172  3.602856  3.517607  3.434374
##  [78]  3.353111  3.273771  3.196308  3.120678  3.046837  2.974744  2.904356
##  [85]  2.835634  2.768538  2.703030  2.639072  2.576627  2.515659  2.456135
##  [92]  2.398018  2.341277  2.285878  2.231791  2.178983  2.127424  2.077086
##  [99]  2.027938  1.979954  1.933105
## 
## 
## [[5]]
## [[5]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[5]]$y
##   [1]  0.000000  4.130090  7.253521  9.593312 11.323571 12.580206 13.469297
##   [8] 14.073631 14.457805 14.672211 14.756149 14.740260 14.648422 14.499231
##  [15] 14.307164 14.083478 13.836919 13.574272 13.300793 13.020541 12.736647
##  [22] 12.451515 12.166983 11.884446 11.604957 11.329302 11.058059 10.791642
##  [29] 10.530344 10.274357 10.023800  9.778735  9.539178  9.305113  9.076497
##  [36]  8.853268  8.635349  8.422655  8.215089  8.012553  7.814944  7.622157
##  [43]  7.434084  7.250620  7.071659  6.897096  6.726827  6.560749  6.398762
##  [50]  6.240768  6.086669  5.936371  5.789781  5.646808  5.507363  5.371360
##  [57]  5.238715  5.109344  4.983167  4.860105  4.740083  4.623023  4.508855
##  [64]  4.397505  4.288906  4.182988  4.079685  3.978934  3.880671  3.784835
##  [71]  3.691365  3.600203  3.511293  3.424578  3.340005  3.257521  3.177074
##  [78]  3.098613  3.022090  2.947456  2.874666  2.803674  2.734435  2.666905
##  [85]  2.601044  2.536808  2.474160  2.413058  2.353465  2.295344  2.238659
##  [92]  2.183373  2.129452  2.076864  2.025574  1.975550  1.926762  1.879179
##  [99]  1.832771  1.787509  1.743365
## 
## 
## [[6]]
## [[6]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[6]]$y
##   [1]  0.000000  4.668450  8.222423 10.903339 12.900865 14.363999 15.409773
##   [8] 16.130100 16.597155 16.867600 16.985905 16.986964 16.898142 16.740890
##  [15] 16.532012 16.284664 16.009130 15.713445 15.403873 15.085290 14.761480
##  [22] 14.435372 14.109223 13.784764 13.463313 13.145863 12.833155 12.525733
##  [29] 12.223984 11.928176 11.638484 11.355008 11.077793 10.806840 10.542117
##  [36] 10.283566 10.031111  9.784661  9.544114  9.309361  9.080290  8.856781
##  [43]  8.638716  8.425975  8.218437  8.015983  7.818494  7.625853  7.437945
##  [50]  7.254657  7.075877  6.901496  6.731407  6.565507  6.403691  6.245862
##  [57]  6.091920  5.941771  5.795322  5.652481  5.513160  5.377272  5.244734
##  [64]  5.115461  4.989375  4.866396  4.746449  4.629457  4.515350  4.404054
##  [71]  4.295502  4.189625  4.086358  3.985637  3.887398  3.791580  3.698124
##  [78]  3.606972  3.518066  3.431352  3.346775  3.264282  3.183823  3.105348
##  [85]  3.028806  2.954151  2.881336  2.810316  2.741047  2.673485  2.607588
##  [92]  2.543315  2.480627  2.419484  2.359847  2.301681  2.244949  2.189614
##  [99]  2.135644  2.083004  2.031662
## 
## 
## [[7]]
## [[7]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[7]]$y
##   [1]  0.000000  4.235390  7.465746  9.907514 11.731010 13.070239 14.030624
##   [8] 14.695093 15.128867 15.383231 15.498500 15.506357 15.431690 15.294041
##  [15] 15.108743 14.887821 14.640691 14.374722 14.095671 13.808027 13.515280
##  [22] 13.220139 12.924695 12.630557 12.338951 12.050808 11.766824 11.487511
##  [29] 11.213239 10.944266 10.680760 10.422826 10.170512  9.923826  9.682747
##  [36]  9.447228  9.217204  8.992597  8.773318  8.559273  8.350360  8.146475
##  [43]  7.947513  7.753367  7.563929  7.379093  7.198751  7.022801  6.851138
##  [50]  6.683661  6.520269  6.360865  6.205354  6.053640  5.905632  5.761241
##  [57]  5.620378  5.482957  5.348896  5.218111  5.090523  4.966055  4.844629
##  [64]  4.726172  4.610611  4.497875  4.387896  4.280606  4.175939  4.073831
##  [71]  3.974220  3.877044  3.782245  3.689763  3.599543  3.511529  3.425667
##  [78]  3.341904  3.260189  3.180473  3.102705  3.026839  2.952829  2.880627
##  [85]  2.810192  2.741478  2.674445  2.609051  2.545255  2.483020  2.422306
##  [92]  2.363077  2.305296  2.248928  2.193939  2.140293  2.087960  2.036906
##  [99]  1.987101  1.938513  1.891113
## 
## 
## [[8]]
## [[8]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[8]]$y
##   [1]  0.000000  4.044547  7.155001  9.528461 11.320807 12.655316 13.629467
##   [8] 14.320340 14.788870 15.083224 15.241471 15.293688 15.263635 15.170071
##  [15] 15.027803 14.848509 14.641397 14.413717 14.171173 13.918246 13.658453
##  [22] 13.394544 13.128666 12.862490 12.597310 12.334121 12.073685 11.816579
##  [29] 11.563234 11.313965 11.068998 10.828488 10.592533 10.361190 10.134481
##  [36]  9.912401  9.694925  9.482014  9.273614  9.069664  8.870098  8.674841
##  [43]  8.483818  8.296951  8.114160  7.935364  7.760482  7.589435  7.422142
##  [50]  7.258524  7.098503  6.942003  6.788946  6.639259  6.492869  6.349703
##  [57]  6.209692  6.072766  5.938858  5.807901  5.679832  5.554585  5.432100
##  [64]  5.312315  5.195171  5.080610  4.968575  4.859011  4.751862  4.647076
##  [71]  4.544601  4.444385  4.346379  4.250535  4.156804  4.065139  3.975496
##  [78]  3.887830  3.802097  3.718255  3.636261  3.556075  3.477658  3.400970
##  [85]  3.325973  3.252630  3.180904  3.110760  3.042162  2.975078  2.909472
##  [92]  2.845314  2.782570  2.721209  2.661202  2.602518  2.545128  2.489004
##  [99]  2.434118  2.380441  2.327949
## 
## 
## [[9]]
## [[9]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[9]]$y
##   [1]  0.000000  4.111367  7.245201  9.611776 11.376639 12.670107 13.594755
##   [8] 14.231307 14.643281 14.880644 14.982695 14.980326 14.897815 14.754231
##  [15] 14.564537 14.340471 14.091228 13.824000 13.544405 13.256825 12.964663
##  [22] 12.670558 12.376546 12.084188 11.794676 11.508907 11.227551 10.951097
##  [29] 10.679897 10.414190 10.154130  9.899806  9.651256  9.408476  9.171434
##  [36]  8.940072  8.714317  8.494082  8.279272  8.069781  7.865503  7.666327
##  [43]  7.472139  7.282828  7.098278  6.918379  6.743017  6.572085  6.405472
##  [50]  6.243073  6.084783  5.930499  5.780123  5.633556  5.490703  5.351469
##  [57]  5.215764  5.083499  4.954587  4.828942  4.706484  4.587130  4.470802
##  [64]  4.357425  4.246922  4.139221  4.034251  3.931943  3.832230  3.735045
##  [71]  3.640324  3.548006  3.458029  3.370334  3.284862  3.201558  3.120367
##  [78]  3.041235  2.964109  2.888939  2.815676  2.744271  2.674676  2.606846
##  [85]  2.540737  2.476304  2.413505  2.352299  2.292644  2.234503  2.177836
##  [92]  2.122606  2.068777  2.016313  1.965179  1.915342  1.866769  1.819428
##  [99]  1.773287  1.728317  1.684487
## 
## 
## [[10]]
## [[10]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[10]]$y
##   [1]  0.000000  4.047895  7.187599  9.603659 11.443530 12.825015 13.842226
##   [8] 14.570333 15.069371 15.387278 15.562323 15.625051 15.599831 15.506095
##  [15] 15.359337 15.171897 14.953600 14.712259 14.454079 14.183985 13.905876
##  [22] 13.622831 13.337278 13.051124 12.765859 12.482642 12.202368 11.925721
##  [29] 11.653217 11.385240 11.122069 10.863896 10.610850 10.363007 10.120399
##  [36]  9.883030  9.650875  9.423891  9.202020  8.985193  8.773331  8.566349
##  [43]  8.364159  8.166667  7.973779  7.785401  7.601435  7.421786  7.246359
##  [50]  7.075060  6.907794  6.744471  6.584999  6.429291  6.277258  6.128815
##  [57]  5.983878  5.842366  5.704197  5.569295  5.437581  5.308981  5.183421
##  [64]  5.060830  4.941137  4.824275  4.710177  4.598776  4.490010  4.383816
##  [71]  4.280134  4.178904  4.080068  3.983569  3.889353  3.797364  3.707552
##  [78]  3.619863  3.534249  3.450659  3.369047  3.289364  3.211567  3.135609
##  [85]  3.061447  2.989040  2.918345  2.849323  2.781932  2.716136  2.651896
##  [92]  2.589175  2.527937  2.468148  2.409773  2.352779  2.297133  2.242802
##  [99]  2.189757  2.137966  2.087401
## 
## 
## [[11]]
## [[11]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[11]]$y
##   [1]  0.000000  3.979585  7.019710  9.322458 11.046854 12.318032 13.234450
##   [8] 13.873562 14.296298 14.550574 14.674065 14.696388 14.640815 14.525623
##  [15] 14.365162 14.170689 13.951027 13.713085 13.462265 13.202786 12.937932
##  [22] 12.670257 12.401738 12.133899 11.867908 11.604654 11.344808 11.088869
##  [29] 10.837200 10.590060 10.347625 10.110007  9.877270  9.649438  9.426503
##  [36]  9.208439  8.995197  8.786720  8.582938  8.383774  8.189147  7.998971
##  [43]  7.813159  7.631622  7.454270  7.281014  7.111765  6.946434  6.784935
##  [50]  6.627180  6.473085  6.322567  6.175545  6.031937  5.891666  5.754655
##  [57]  5.620827  5.490111  5.362434  5.237724  5.115915  4.996937  4.880727
##  [64]  4.767218  4.656349  4.548058  4.442286  4.338973  4.238063  4.139500
##  [71]  4.043228  3.949196  3.857351  3.767641  3.680018  3.594433  3.510838
##  [78]  3.429187  3.349435  3.271538  3.195453  3.121137  3.048550  2.977650
##  [85]  2.908400  2.840760  2.774693  2.710162  2.647133  2.585569  2.525437
##  [92]  2.466703  2.409336  2.353302  2.298572  2.245115  2.192901  2.141901
##  [99]  2.092087  2.043432  1.995908
## 
## 
## [[12]]
## [[12]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[12]]$y
##   [1]  0.000000  4.235348  7.469252  9.916409 11.745977 13.091219 14.057110
##   [8] 14.726330 15.163989 15.421347 15.538755 15.547961 15.473937 15.336314
##  [15] 15.150517 14.928653 14.680221 14.412660 14.131787 13.842148 13.547279
##  [22] 13.249928 12.952219 12.655790 12.361889 12.071465 11.785230 11.503707
##  [29] 11.227276 10.956201 10.690658 10.430752 10.176536  9.928020  9.685183
##  [36]  9.447977  9.216339  8.990188  8.769437  8.553987  8.343736  8.138580
##  [43]  7.938410  7.743118  7.552594  7.366729  7.185416  7.008548  6.836019
##  [50]  6.667727  6.503569  6.343446  6.187260  6.034915  5.886318  5.741377
##  [57]  5.600003  5.462108  5.327608  5.196419  5.068459  4.943650  4.821913
##  [64]  4.703174  4.587359  4.474395  4.364213  4.256743  4.151921  4.049679
##  [71]  3.949955  3.852687  3.757814  3.665277  3.575019  3.486983  3.401115
##  [78]  3.317362  3.235671  3.155992  3.078275  3.002472  2.928535  2.856419
##  [85]  2.786079  2.717472  2.650553  2.585283  2.521620  2.459524  2.398958
##  [92]  2.339883  2.282263  2.226061  2.171244  2.117777  2.065626  2.014759
##  [99]  1.965146  1.916753  1.869553
## 
## 
## [[13]]
## [[13]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[13]]$y
##   [1]  0.000000  4.468826  7.850710 10.386041 12.262571 13.626930 14.593620
##   [8] 15.252041 15.671984 15.907920 16.002350 15.988426 15.891995 15.733201
##  [15] 15.527729 15.287785 15.022856 14.740307 14.445847 14.143888 13.837835
##  [22] 13.530305 13.223300 12.918343 12.616583 12.318879 12.025866 11.737999
##  [29] 11.455600 11.178885 10.907985 10.642973 10.383871 10.130662  9.883305
##  [36]  9.641736  9.405873  9.175627  8.950897  8.731576  8.517556  8.308725
##  [43]  8.104969  7.906175  7.712229  7.523019  7.338435  7.158367  6.982707
##  [50]  6.811349  6.644191  6.481130  6.322067  6.166904  6.015548  5.867905
##  [57]  5.723883  5.583396  5.446356  5.312678  5.182282  5.055085  4.931010
##  [64]  4.809980  4.691921  4.576759  4.464423  4.354845  4.247956  4.143691
##  [71]  4.041985  3.942775  3.846001  3.751601  3.659519  3.569696  3.482079
##  [78]  3.396612  3.313242  3.231919  3.152592  3.075212  2.999732  2.926104
##  [85]  2.854283  2.784225  2.715887  2.649226  2.584201  2.520772  2.458900
##  [92]  2.398547  2.339675  2.282248  2.226230  2.171588  2.118287  2.066293
##  [99]  2.015577  1.966105  1.917847
## 
## 
## [[14]]
## [[14]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[14]]$y
##   [1]  0.000000  4.123353  7.258186  9.621008 11.381333 12.671871 13.596505
##   [8] 14.236547 14.655635 14.903560 15.019276 15.033247 14.969291 14.846020
##  [15] 14.677971 14.476493 14.250433 14.006685 13.750612 13.486380 13.217215
##  [22] 12.945612 12.673493 12.402330 12.133244 11.867085 11.604486 11.345916
##  [29] 11.091713 10.842114 10.597278 10.357301 10.122235  9.892094  9.666864
##  [36]  9.446512  9.230985  9.020222  8.814150  8.612691  8.415763  8.223279
##  [43]  8.035153  7.851295  7.671616  7.496027  7.324440  7.156768  6.992923
##  [50]  6.832821  6.676378  6.523513  6.374143  6.228190  6.085577  5.946227
##  [57]  5.810067  5.677024  5.547026  5.420004  5.295891  5.174619  5.056124
##  [64]  4.940342  4.827211  4.716670  4.608661  4.503125  4.400006  4.299248
##  [71]  4.200797  4.104601  4.010607  3.918766  3.829028  3.741345  3.655670
##  [78]  3.571956  3.490160  3.410237  3.332144  3.255839  3.181282  3.108432
##  [85]  3.037250  2.967699  2.899739  2.833337  2.768454  2.705058  2.643113
##  [92]  2.582587  2.523447  2.465661  2.409199  2.354029  2.300123  2.247451
##  [99]  2.195985  2.145698  2.096562
## 
## 
## [[15]]
## [[15]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[15]]$y
##   [1]  0.000000  4.295524  7.536872  9.960281 11.749526 13.047562 13.965577
##   [8] 14.590043 14.988197 15.212312 15.303017 15.291881 15.203420 15.056671
##  [15] 14.866401 14.644058 14.398508 14.136610 13.863661 13.583742 13.299993
##  [22] 13.014817 12.730049 12.447080 12.166960 11.890469 11.618182 11.350513
##  [29] 11.087755 10.830101 10.577674 10.330539 10.088718  9.852200  9.620948
##  [36]  9.394907  9.174008  8.958172  8.747312  8.541335  8.340148  8.143650
##  [43]  7.951745  7.764333  7.581316  7.402595  7.228073  7.057655  6.891247
##  [50]  6.728756  6.570091  6.415164  6.263887  6.116174  5.971944  5.831113
##  [57]  5.693602  5.559333  5.428230  5.300218  5.175224  5.053178  4.934009
##  [64]  4.817651  4.704037  4.593102  4.484783  4.379019  4.275748  4.174913
##  [71]  4.076456  3.980321  3.886453  3.794799  3.705306  3.617924  3.532602
##  [78]  3.449293  3.367948  3.288522  3.210968  3.135244  3.061305  2.989111
##  [85]  2.918618  2.849788  2.782582  2.716960  2.652886  2.590323  2.529235
##  [92]  2.469588  2.411348  2.354481  2.298955  2.244739  2.191801  2.140112
##  [99]  2.089641  2.040361  1.992243
## 
## 
## [[16]]
## [[16]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[16]]$y
##   [1]  0.000000  4.528854  7.972500 10.568405 12.502543 13.920551 14.936464
##   [8] 15.639565 16.099748 16.371717 16.498279 16.512920 16.441829 16.305475
##  [15] 16.119852 15.897445 15.647992 15.379084 15.096626 14.805204 14.508374
##  [22] 14.208884 13.908852 13.609899 13.313263 13.019878 12.730443 12.445474
##  [29] 12.165344 11.890313 11.620555 11.356178 11.097237 10.843746 10.595691
##  [36] 10.353031 10.115712  9.883661  9.656800  9.435041  9.218293  9.006459
##  [43]  8.799444  8.597147  8.399469  8.206313  8.017580  7.833173  7.652995
##  [50]  7.476953  7.304953  7.136904  6.972717  6.812303  6.655578  6.502455
##  [57]  6.352854  6.206694  6.063895  5.924381  5.788076  5.654907  5.524801
##  [64]  5.397688  5.273500  5.152168  5.033628  4.917816  4.804668  4.694123
##  [71]  4.586121  4.480604  4.377515  4.276798  4.178398  4.082262  3.988338
##  [78]  3.896575  3.806923  3.719334  3.633760  3.550155  3.468473  3.388671
##  [85]  3.310705  3.234532  3.160113  3.087405  3.016371  2.946970  2.879167
##  [92]  2.812923  2.748204  2.684974  2.623198  2.562844  2.503878  2.446269
##  [99]  2.389986  2.334997  2.281274
## 
## 
## [[17]]
## [[17]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[17]]$y
##   [1]  0.000000  4.678595  8.149023 10.695552 12.536201 13.838163 14.729639
##   [8] 15.308916 15.651331 15.814618 15.843006 15.770361 15.622606 15.419567
##  [15] 15.176396 14.904659 14.613173 14.308648 13.996179 13.679622 13.361882
##  [22] 13.045142 12.731026 12.420732 12.115135 11.814861 11.520345 11.231881
##  [29] 10.949649 10.673750 10.404221 10.141051  9.884195  9.633583  9.389123
##  [36]  9.150712  8.918236  8.691575  8.470605  8.255199  8.045230  7.840570
##  [43]  7.641092  7.446670  7.257181  7.072503  6.892516  6.717104  6.546150
##  [50]  6.379544  6.217175  6.058936  5.904723  5.754434  5.607969  5.465231
##  [57]  5.326125  5.190560  5.058445  4.929692  4.804217  4.681935  4.562765
##  [64]  4.446628  4.333448  4.223148  4.115655  4.010899  3.908809  3.809317
##  [71]  3.712358  3.617867  3.525780  3.436038  3.348580  3.263348  3.180285
##  [78]  3.099337  3.020449  2.943569  2.868645  2.795629  2.724472  2.655125
##  [85]  2.587544  2.521682  2.457497  2.394946  2.333987  2.274580  2.216685
##  [92]  2.160263  2.105277  2.051691  1.999469  1.948576  1.898979  1.850644
##  [99]  1.803539  1.757633  1.712896
## 
## 
## [[18]]
## [[18]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[18]]$y
##   [1]  0.000000  4.353980  7.657212 10.141819 11.989081 13.340565 14.306814
##   [8] 14.974136 15.409910 15.666727 15.785627 15.798628 15.730701 15.601312
##  [15] 15.425629 15.215461 14.979997 14.726376 14.460141 14.185586 13.906031
##  [22] 13.624035 13.341565 13.060125 12.780859 12.504629 12.232078 11.963680
##  [29] 11.699775 11.440602 11.186320 10.937025 10.692767 10.453559 10.219386
##  [36]  9.990211  9.765983  9.546636  9.332099  9.122290  8.917127  8.716521
##  [43]  8.520385  8.328628  8.141160  7.957890  7.778730  7.603590  7.432384
##  [50]  7.265025  7.101429  6.941512  6.785192  6.632389  6.483026  6.337024
##  [57]  6.194309  6.054807  5.918446  5.785155  5.654865  5.527510  5.403022
##  [64]  5.281338  5.162394  5.046129  4.932482  4.821394  4.712809  4.606668
##  [71]  4.502919  4.401505  4.302376  4.205479  4.110765  4.018184  3.927687
##  [78]  3.839229  3.752763  3.668245  3.585630  3.504875  3.425939  3.348781
##  [85]  3.273361  3.199640  3.127578  3.057140  2.988288  2.920987  2.855201
##  [92]  2.790897  2.728041  2.666601  2.606545  2.547841  2.490459  2.434370
##  [99]  2.379544  2.325952  2.273568
## 
## 
## [[19]]
## [[19]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[19]]$y
##   [1]  0.000000  4.058248  7.143729  9.466537 11.191939 12.449924 13.342720
##   [8] 13.950709 14.337082 14.551504 14.632994 14.612197 14.513167 14.354775
##  [15] 14.151814 13.915867 13.655996 13.379277 13.091226 12.796134 12.497326
##  [22] 12.197371 11.898244 11.601452 11.308136 11.019150 10.735126 10.456517
##  [29] 10.183640  9.916707  9.655843  9.401112  9.152528  8.910066  8.673672
##  [36]  8.443271  8.218770  8.000065  7.787044  7.579589  7.377575  7.180880
##  [43]  6.989377  6.802939  6.621443  6.444763  6.272777  6.105365  5.942409
##  [50]  5.783793  5.629403  5.479128  5.332859  5.190492  5.051922  4.917050
##  [57]  4.785776  4.658006  4.533645  4.412604  4.294794  4.180129  4.068525
##  [64]  3.959900  3.854174  3.751272  3.651117  3.553635  3.458756  3.366411
##  [71]  3.276530  3.189050  3.103905  3.021033  2.940374  2.861868  2.785459
##  [78]  2.711089  2.638705  2.568254  2.499684  2.432944  2.367987  2.304763
##  [85]  2.243228  2.183335  2.125042  2.068305  2.013083  1.959335  1.907023
##  [92]  1.856107  1.806550  1.758316  1.711371  1.665679  1.621206  1.577921
##  [99]  1.535792  1.494788  1.454878
## 
## 
## [[20]]
## [[20]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[20]]$y
##   [1]  0.000000  4.039666  7.149394  9.523453 11.315947 12.649116 13.619909
##   [8] 14.305209 14.765977 15.050542 15.197206 15.236319 15.191922 15.083045
##  [15] 14.924751 14.728948 14.505049 14.260479 14.001096 13.731509 13.455337
##  [22] 13.175420 12.893972 12.612719 12.332996 12.055829 11.782001 11.512103
##  [29] 11.246572 10.985727 10.729790 10.478910 10.233178  9.992639  9.757303
##  [36]  9.527150  9.302142  9.082222  8.867323  8.657366  8.452268  8.251939
##  [43]  8.056288  7.865219  7.678636  7.496445  7.318548  7.144851  6.975258
##  [50]  6.809677  6.648015  6.490182  6.336090  6.185650  6.038778  5.895390
##  [57]  5.755403  5.618739  5.485317  5.355063  5.227900  5.103757  4.982560
##  [64]  4.864241  4.748731  4.635964  4.525875  4.418399  4.313476  4.211044
##  [71]  4.111044  4.013419  3.918112  3.825068  3.734234  3.645557  3.558986
##  [78]  3.474470  3.391961  3.311412  3.232776  3.156007  3.081061  3.007894
##  [85]  2.936466  2.866733  2.798656  2.732196  2.667315  2.603974  2.542137
##  [92]  2.481768  2.422833  2.365298  2.309129  2.254294  2.200761  2.148499
##  [99]  2.097479  2.047669  1.999043
## 
## 
## [[21]]
## [[21]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[21]]$y
##   [1]  0.000000  4.465892  7.818500 10.313304 12.147592 13.473705 14.409269
##   [8] 15.045096 15.451288 15.681950 15.778835 15.774152 15.692742 15.553757
##  [15] 15.371949 15.158681 14.922693 14.670701 14.407859 14.138117 13.864492
##  [22] 13.589283 13.314237 13.040671 12.769574 12.501681 12.237532 11.977516
##  [29] 11.721906 11.470888 11.224579 10.983044 10.746309 10.514370 10.287199
##  [36] 10.064754  9.846978  9.633804  9.425161  9.220972  9.021156  8.825630
##  [43]  8.634313  8.447120  8.263967  8.084772  7.909452  7.737925  7.570111
##  [50]  7.405932  7.245310  7.088169  6.934434  6.784031  6.636889  6.492937
##  [57]  6.352107  6.214331  6.079542  5.947677  5.818672  5.692464  5.568994
##  [64]  5.448202  5.330030  5.214420  5.101319  4.990670  4.882421  4.776521
##  [71]  4.672917  4.571561  4.472402  4.375395  4.280492  4.187647  4.096816
##  [78]  4.007955  3.921022  3.835974  3.752771  3.671372  3.591740  3.513834
##  [85]  3.437618  3.363055  3.290110  3.218747  3.148931  3.080630  3.013811
##  [92]  2.948441  2.884488  2.821923  2.760715  2.700835  2.642253  2.584942
##  [99]  2.528874  2.474022  2.420360
## 
## 
## [[22]]
## [[22]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[22]]$y
##   [1]  0.000000  4.453198  7.818476 10.335998 12.193512 13.537776 14.483494
##   [8] 15.120296 15.518212 15.731938 15.804183 15.768282 15.650238 15.470327
##  [15] 15.244341 14.984571 14.700571 14.399754 14.087863 13.769334 13.447581
##  [22] 13.125226 12.804266 12.486215 12.172208 11.863087 11.559464 11.261775
##  [29] 10.970316 10.685279 10.406770 10.134834  9.869468  9.610631  9.358255
##  [36]  9.112250  8.872511  8.638923  8.411362  8.189699  7.973803  7.763540
##  [43]  7.558775  7.359375  7.165207  6.976140  6.792045  6.612795  6.438265
##  [50]  6.268333  6.102879  5.941788  5.784945  5.632239  5.483562  5.338807
##  [57]  5.197873  5.060657  4.927063  4.796995  4.670360  4.547067  4.427029
##  [64]  4.310160  4.196376  4.085595  3.977739  3.872730  3.770493  3.670955
##  [71]  3.574044  3.479692  3.387831  3.298395  3.211319  3.126543  3.044004
##  [78]  2.963645  2.885407  2.809234  2.735072  2.662868  2.592570  2.524128
##  [85]  2.457493  2.392617  2.329454  2.267958  2.208085  2.149793  2.093040
##  [92]  2.037786  1.983990  1.931614  1.880620  1.830973  1.782637  1.735577
##  [99]  1.689759  1.645150  1.601719
## 
## 
## [[23]]
## [[23]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[23]]$y
##   [1]  0.000000  4.303527  7.620898 10.156560 12.073044 13.499549 14.538765
##   [8] 15.272303 15.765010 16.068402 16.223395 16.262478 16.211435 16.090724
##  [15] 15.916565 15.701809 15.456630 15.189074 14.905496 14.610906 14.309245
##  [22] 14.003609 13.696417 13.389555 13.084484 12.782329 12.483949 12.189990
##  [29] 11.900932 11.617122 11.338805 11.066141 10.799227 10.538111 10.282798
##  [36] 10.033266  9.789467  9.551337  9.318795  9.091752  8.870113  8.653774
##  [43]  8.442630  8.236574  8.035494  7.839283  7.647831  7.461028  7.278767
##  [50]  7.100942  6.927449  6.758184  6.593047  6.431938  6.274761  6.121420
##  [57]  5.971824  5.825881  5.683502  5.544602  5.409095  5.276898  5.147931
##  [64]  5.022116  4.899375  4.779633  4.662818  4.548857  4.437682  4.329223
##  [71]  4.223415  4.120193  4.019494  3.921255  3.825418  3.731923  3.640713
##  [78]  3.551732  3.464926  3.380241  3.297626  3.217030  3.138404  3.061700
##  [85]  2.986871  2.913870  2.842653  2.773177  2.705399  2.639278  2.574772
##  [92]  2.511844  2.450453  2.390562  2.332136  2.275137  2.219531  2.165285
##  [99]  2.112364  2.060737  2.010371
## 
## 
## [[24]]
## [[24]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[24]]$y
##   [1]  0.000000  4.131721  7.299590  9.707658 11.517227 12.855795 13.824128
##   [8] 14.501857 14.951902 15.223964 15.357298 15.382898 15.325225 15.203575
##  [15] 15.033165 14.825980 14.591458 14.337019 14.068489 13.790437 13.506434
##  [22] 13.219270 12.931111 12.643634 12.358131 12.075589 11.796754 11.522182
##  [29] 11.252284 10.987351 10.727583 10.473109 10.224002  9.980290  9.741968
##  [36]  9.509007  9.281355  9.058945  8.841702  8.629539  8.422365  8.220084
##  [43]  8.022597  7.829804  7.641604  7.457896  7.278579  7.103554  6.932723
##  [50]  6.765987  6.603251  6.444422  6.289407  6.138116  5.990461  5.846354
##  [57]  5.705712  5.568451  5.434491  5.303752  5.176157  5.051631  4.930101
##  [64]  4.811494  4.695739  4.582770  4.472518  4.364918  4.259907  4.157421
##  [71]  4.057402  3.959789  3.864523  3.771550  3.680814  3.592260  3.505837
##  [78]  3.421493  3.339178  3.258844  3.180442  3.103927  3.029252  2.956374
##  [85]  2.885249  2.815835  2.748091  2.681977  2.617454  2.554482  2.493026
##  [92]  2.433049  2.374514  2.317387  2.261635  2.207224  2.154123  2.102298
##  [99]  2.051721  2.002360  1.954187
## 
## 
## [[25]]
## [[25]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[25]]$y
##   [1]  0.000000  4.224211  7.469795  9.942571 11.805520 13.187686 14.191234
##   [8] 14.897043 15.369123 15.658131 15.804144 15.838859 15.787340 15.669391
##  [15] 15.500658 15.293489 15.057625 14.800737 14.528863 14.246746 13.958104
##  [22] 13.665845 13.372234 13.079031 12.787594 12.498965 12.213935 11.933097
##  [29] 11.656892 11.385635 11.119547 10.858772 10.603396 10.353458 10.108962
##  [36]  9.869883  9.636177  9.407782  9.184623  8.966617  8.753675  8.545702
##  [43]  8.342601  8.144271  7.950613  7.761525  7.576907  7.396658  7.220680
##  [50]  7.048876  6.881148  6.717403  6.557547  6.401491  6.249144  6.100419
##  [57]  5.955231  5.813496  5.675133  5.540062  5.408205  5.279485  5.153827
##  [64]  5.031161  4.911413  4.794515  4.680400  4.569000  4.460251  4.354091
##  [71]  4.250457  4.149290  4.050531  3.954122  3.860008  3.768134  3.678447
##  [78]  3.590894  3.505426  3.421991  3.340542  3.261032  3.183415  3.107645
##  [85]  3.033678  2.961472  2.890984  2.822174  2.755002  2.689429  2.625416
##  [92]  2.562927  2.501926  2.442376  2.384244  2.327495  2.272097  2.218018
##  [99]  2.165225  2.113690  2.063381
## 
## 
## [[26]]
## [[26]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[26]]$y
##   [1]  0.000000  4.790887  8.392196 11.073936 13.045369 14.468668 15.469496
##   [8] 16.145220 16.571263 16.806044 16.894798 16.872540 16.766365 16.597227
##  [15] 16.381323 16.131159 15.856385 15.564437 15.261033 14.950564 14.636390
##  [22] 14.321075 14.006565 13.694330 13.385469 13.080799 12.780914 12.486238
##  [29] 12.197066 11.913592 11.635931 11.364140 11.098233 10.838187 10.583955
##  [36] 10.335469 10.092649  9.855404  9.623635  9.397238  9.176106  8.960131
##  [43]  8.749203  8.543211  8.342048  8.145605  7.953774  7.766451  7.583531
##  [50]  7.404914  7.230499  7.060188  6.893886  6.731499  6.572936  6.418106
##  [57]  6.266922  6.119299  5.975152  5.834401  5.696964  5.562765  5.431727
##  [64]  5.303776  5.178838  5.056844  4.937723  4.821408  4.707833  4.596933
##  [71]  4.488646  4.382909  4.279664  4.178850  4.080411  3.984291  3.890435
##  [78]  3.798791  3.709305  3.621927  3.536607  3.453297  3.371950  3.292519
##  [85]  3.214959  3.139226  3.065277  2.993070  2.922563  2.853718  2.786495
##  [92]  2.720855  2.656761  2.594177  2.533068  2.473398  2.415133  2.358241
##  [99]  2.302690  2.248446  2.195481
## 
## 
## [[27]]
## [[27]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[27]]$y
##   [1]  0.000000  4.054127  7.131391  9.445735 11.164715 12.419551 13.312989
##   [8] 13.925477 14.319985 14.545799 14.641479 14.637189 14.556511 14.417873
##  [15] 14.235668 14.021127 13.783005 13.528118 13.261764 12.988056 12.710172
##  [22] 12.430568 12.151126 11.873286 11.598139 11.326507 11.058996 10.796051
##  [29] 10.537986 10.285014 10.037275  9.794844  9.557752  9.325995  9.099541
##  [36]  8.878338  8.662318  8.451402  8.245502  8.044526  7.848374  7.656948
##  [43]  7.470146  7.287866  7.110007  6.936466  6.767145  6.601943  6.440764
##  [50]  6.283512  6.130093  5.980414  5.834387  5.691922  5.552933  5.417336
##  [57]  5.285049  5.155992  5.030084  4.907251  4.787417  4.670508  4.556455
##  [64]  4.445186  4.336634  4.230732  4.127417  4.026625  3.928294  3.832364
##  [71]  3.738777  3.647475  3.558402  3.471505  3.386730  3.304025  3.223340
##  [78]  3.144625  3.067832  2.992915  2.919827  2.848524  2.778962  2.711099
##  [85]  2.644893  2.580304  2.517292  2.455819  2.395847  2.337340  2.280262
##  [92]  2.224577  2.170252  2.117254  2.065550  2.015109  1.965899  1.917891
##  [99]  1.871056  1.825364  1.780788
## 
## 
## [[28]]
## [[28]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[28]]$y
##   [1]  0.000000  4.228019  7.440256  9.859331 11.659526 12.977269 13.919350
##   [8] 14.569344 14.992653 15.240443 15.352739 15.360838 15.289204 15.156952
##  [15] 14.979008 14.767020 14.530069 14.275226 14.007990 13.732630 13.452451
##  [22] 13.170007 12.887261 12.605716 12.326518 12.050528 11.778390 11.510578
##  [29] 11.247430 10.989183 10.735994 10.487954 10.245111 10.007472  9.775017
##  [36]  9.547705  9.325478  9.108266  8.895990  8.688565  8.485901  8.287905
##  [43]  8.094482  7.905538  7.720975  7.540699  7.364614  7.192628  7.024647
##  [50]  6.860581  6.700341  6.543838  6.390987  6.241702  6.095903  5.953507
##  [57]  5.814436  5.678612  5.545960  5.416406  5.289878  5.166306  5.045619
##  [64]  4.927752  4.812638  4.700212  4.590413  4.483179  4.378450  4.276167
##  [71]  4.176274  4.078714  3.983433  3.890378  3.799496  3.710738  3.624053
##  [78]  3.539393  3.456711  3.375961  3.297096  3.220074  3.144852  3.071386
##  [85]  2.999637  2.929564  2.861127  2.794290  2.729014  2.665262  2.603000
##  [92]  2.542193  2.482806  2.424806  2.368161  2.312840  2.258810  2.206043
##  [99]  2.154509  2.104178  2.055024
## 
## 
## [[29]]
## [[29]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[29]]$y
##   [1]  0.000000  4.195755  7.350440  9.700569 11.429385 12.678839 13.558892
##   [8] 14.154709 14.532245 14.742564 14.825197 14.810731 14.722832 14.579793
##  [15] 14.395755 14.181632 13.945846 13.694884 13.433737 13.166236 12.895314
##  [22] 12.623210 12.351624 12.081843 11.814828 11.551296 11.291769 11.036623
##  [29] 10.786119 10.540431 10.299666 10.063878  9.833083  9.607268  9.386396
##  [36]  9.170413  8.959255  8.752846  8.551104  8.353945  8.161279  7.973015
##  [43]  7.789063  7.609330  7.433726  7.262160  7.094542  6.930783  6.770798
##  [50]  6.614501  6.461807  6.312636  6.166905  6.024537  5.885454  5.749580
##  [57]  5.616843  5.487169  5.360489  5.236732  5.115833  4.997724  4.882343
##  [64]  4.769624  4.659508  4.551934  4.446844  4.344179  4.243885  4.145907
##  [71]  4.050190  3.956683  3.865335  3.776096  3.688917  3.603751  3.520551
##  [78]  3.439272  3.359869  3.282300  3.206521  3.132492  3.060172  2.989522
##  [85]  2.920503  2.853077  2.787208  2.722859  2.659997  2.598585  2.538591
##  [92]  2.479983  2.422727  2.366794  2.312151  2.258771  2.206622  2.155678
##  [99]  2.105910  2.057291  2.009794
## 
## 
## [[30]]
## [[30]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[30]]$y
##   [1]  0.000000  4.392928  7.672879 10.097157 11.864130 13.126713 14.002772
##   [8] 14.583143 14.937820 15.120725 15.173384 15.127770 15.008483 14.834447
##  [15] 14.620202 14.376913 14.113142 13.835444 13.548830 13.257119 12.963211
##  [22] 12.669302 12.377043 12.087667 11.802085 11.520962 11.244770 10.973842
##  [29] 10.708393 10.448558 10.194405  9.945954  9.703189  9.466064  9.234513
##  [36]  9.008457  8.787804  8.572455  8.362305  8.157247  7.957171  7.761967
##  [43]  7.571524  7.385732  7.204483  7.027669  6.855185  6.686927  6.522793
##  [50]  6.362683  6.206500  6.054148  5.905534  5.760566  5.619156  5.481216
##  [57]  5.346661  5.215409  5.087379  4.962491  4.840669  4.721837  4.605922
##  [64]  4.492853  4.382559  4.274973  4.170027  4.067659  3.967803  3.870398
##  [71]  3.775384  3.682703  3.592297  3.504111  3.418089  3.334179  3.252329
##  [78]  3.172488  3.094608  3.018639  2.944535  2.872250  2.801740  2.732961
##  [85]  2.665870  2.600426  2.536588  2.474318  2.413577  2.354326  2.296530
##  [92]  2.240153  2.185160  2.131517  2.079191  2.028149  1.978361  1.929794
##  [99]  1.882420  1.836209  1.791133
## 
## 
## [[31]]
## [[31]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[31]]$y
##   [1]  0.000000  4.371758  7.736771 10.306628 12.248856 13.696087 14.753314
##   [8] 15.503646 16.012860 16.333006 16.505266 16.562212 16.529601 16.427794
##  [15] 16.272875 16.077548 15.851835 15.603638 15.339180 15.063356 14.780007
##  [22] 14.492145 14.202120 13.911763 13.622495 13.335410 13.051344 12.770932
##  [29] 12.494648 12.222841 11.955758 11.693571 11.436389 11.184274 10.937251
##  [36] 10.695314 10.458436 10.226572  9.999666  9.777649  9.560446  9.347977
##  [43]  9.140158  8.936902  8.738120  8.543724  8.353625  8.167733  7.985959
##  [50]  7.808217  7.634420  7.464482  7.298320  7.135851  6.976994  6.821671
##  [57]  6.669803  6.521313  6.376127  6.234173  6.095378  5.959672  5.826986
##  [64]  5.697254  5.570410  5.446390  5.325131  5.206571  5.090651  4.977311
##  [71]  4.866495  4.758146  4.652209  4.548631  4.447359  4.348341  4.251528
##  [78]  4.156871  4.064321  3.973831  3.885357  3.798852  3.714273  3.631577
##  [85]  3.550722  3.471667  3.394373  3.318799  3.244909  3.172663  3.102025
##  [92]  3.032961  2.965434  2.899410  2.834857  2.771741  2.710029  2.649692
##  [99]  2.590699  2.533018  2.476622
## 
## 
## [[32]]
## [[32]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[32]]$y
##   [1]  0.000000  4.092929  7.241412  9.643696 11.456828 12.805202 13.787343
##   [8] 14.481275 14.948779 15.238767 15.389952 15.432969 15.392051 15.286359
##  [15] 15.131036 14.938044 14.716820 14.474809 14.217869 13.950611 13.676650
##  [22] 13.398818 13.119327 12.839894 12.561851 12.286221 12.013783 11.745128
##  [29] 11.480692 11.220793 10.965656 10.715433 10.470215 10.230053  9.994957
##  [36]  9.764915  9.539890  9.319831  9.104674  8.894347  8.688769  8.487857
##  [43]  8.291521  8.099673  7.912220  7.729072  7.550136  7.375321  7.204537
##  [50]  7.037695  6.874705  6.715481  6.559939  6.407994  6.259564  6.114569
##  [57]  5.972930  5.834570  5.699413  5.567386  5.438416  5.312433  5.189368
##  [64]  5.069153  4.951723  4.837012  4.724959  4.615501  4.508579  4.404134
##  [71]  4.302108  4.202445  4.105092  4.009993  3.917098  3.826354  3.737713
##  [78]  3.651125  3.566543  3.483920  3.403212  3.324373  3.247360  3.172132
##  [85]  3.098646  3.026863  2.956742  2.888246  2.821337  2.755978  2.692133
##  [92]  2.629767  2.568845  2.509335  2.451204  2.394419  2.338950  2.284766
##  [99]  2.231837  2.180134  2.129629
## 
## 
## [[33]]
## [[33]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[33]]$y
##   [1]  0.000000  4.387792  7.733312 10.262768 12.153710 13.545488 14.547461
##   [8] 15.245445 15.706768 15.984245 16.119300 16.144413 16.085040 15.961128
##  [15] 15.788298 15.578772 15.342109 15.085776 14.815598 14.536112 14.250846
##  [22] 13.962533 13.673288 13.384736 13.098121 12.814389 12.534252 12.258236
##  [29] 11.986729 11.720003 11.458246 11.201576 10.950060 10.703724 10.462563
##  [36] 10.226547  9.995629  9.769748  9.548832  9.332802  9.121574  8.915059
##  [43]  8.713166  8.515804  8.322880  8.134301  7.949975  7.769809  7.593715
##  [50]  7.421601  7.253381  7.088968  6.928277  6.771224  6.617729  6.467712
##  [57]  6.321093  6.177796  6.037747  5.900872  5.767099  5.636358  5.508581
##  [64]  5.383700  5.261650  5.142367  5.025787  4.911851  4.800497  4.691668
##  [71]  4.585306  4.481355  4.379761  4.280469  4.183429  4.088589  3.995899
##  [78]  3.905310  3.816774  3.730246  3.645680  3.563030  3.482255  3.403310
##  [85]  3.326156  3.250750  3.177054  3.105029  3.034636  2.965840  2.898603
##  [92]  2.832890  2.768667  2.705900  2.644556  2.584602  2.526008  2.468742
##  [99]  2.412775  2.358076  2.304617
## 
## 
## [[34]]
## [[34]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[34]]$y
##   [1]  0.000000  4.151702  7.281642  9.617623 11.337207 12.578781 13.450187
##   [8] 14.035450 14.400026 14.594892 14.659737 14.625450 14.516058 14.350241
##  [15] 14.142508 13.904121 13.643807 13.368321 13.082881 12.791506 12.497284
##  [22] 12.202575 11.909177 11.618446 11.331398 11.048783 10.771146 10.498872
##  [29] 10.232224  9.971370  9.716401  9.467358  9.224233  8.986991  8.755569
##  [36]  8.529886  8.309849  8.095354  7.886292  7.682547  7.484003  7.290541
##  [43]  7.102041  6.918385  6.739455  6.565134  6.395308  6.229864  6.068691
##  [50]  5.911681  5.758728  5.609729  5.464581  5.323186  5.185448  5.051273
##  [57]  4.920568  4.793244  4.669214  4.548393  4.430698  4.316048  4.204365
##  [64]  4.095571  3.989592  3.886356  3.785790  3.687827  3.592399  3.499441
##  [71]  3.408887  3.320677  3.234749  3.151045  3.069507  2.990079  2.912706
##  [78]  2.837335  2.763914  2.692394  2.622724  2.554857  2.488746  2.424346
##  [85]  2.361612  2.300501  2.240972  2.182984  2.126495  2.071469  2.017866
##  [92]  1.965651  1.914787  1.865239  1.816973  1.769956  1.724155  1.679540
##  [99]  1.636079  1.593743  1.552502
## 
## 
## [[35]]
## [[35]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[35]]$y
##   [1]  0.000000  4.683663  8.177987 10.758976 12.639153 13.982145 14.913920
##   [8] 15.531441 15.909335 16.105022 16.162671 16.116248 15.991860 15.809564
##  [15] 15.584754 15.329239 15.052064 14.760149 14.458774 14.151965 13.842773
##  [22] 13.533508 13.225906 12.921261 12.620531 12.324414 12.033407 11.747858
##  [29] 11.467997 11.193965 10.925835 10.663632 10.407339 10.156910  9.912281
##  [36]  9.673370  9.440081  9.212316  8.989966  8.772921  8.561069  8.354296
##  [43]  8.152490  7.955536  7.763324  7.575744  7.392685  7.214043  7.039711
##  [50]  6.869588  6.703573  6.541566  6.383474  6.229200  6.078653  5.931744
##  [57]  5.788385  5.648489  5.511975  5.378759  5.248763  5.121908  4.998119
##  [64]  4.877322  4.759444  4.644415  4.532166  4.422630  4.315742  4.211436
##  [71]  4.109652  4.010327  3.913403  3.818821  3.726526  3.636461  3.548573
##  [78]  3.462808  3.379117  3.297449  3.217754  3.139985  3.064096  2.990041
##  [85]  2.917776  2.847257  2.778443  2.711292  2.645764  2.581819  2.519420
##  [92]  2.458529  2.399110  2.341127  2.284545  2.229331  2.175451  2.122873
##  [99]  2.071566  2.021499  1.972642
## 
## 
## [[36]]
## [[36]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[36]]$y
##   [1]  0.000000  4.638622  8.115532 10.697236 12.589618 13.951714 14.906361
##   [8] 15.548427 15.951168 16.171137 16.251988 16.227404 16.123362 15.959894
##  [15] 15.752429 15.512848 15.250287 14.971763 14.682657 14.387087 14.088194
##  [22] 13.788367 13.489416 13.192699 12.899233 12.609766 12.324843 12.044850
##  [29] 11.770055 11.500629 11.236676 10.978245 10.725343 10.477946 10.236009
##  [36]  9.999467  9.768243  9.542252  9.321401  9.105594  8.894730  8.688709
##  [43]  8.487428  8.290786  8.098680  7.911012  7.727681  7.548590  7.373643
##  [50]  7.202745  7.035804  6.872729  6.713432  6.557825  6.405823  6.257344
##  [57]  6.112305  5.970627  5.832233  5.697046  5.564993  5.436000  5.309997
##  [64]  5.186914  5.066684  4.949241  4.834521  4.722459  4.612995  4.506068
##  [71]  4.401620  4.299593  4.199930  4.102578  4.007482  3.914591  3.823853
##  [78]  3.735218  3.648637  3.564064  3.481450  3.400752  3.321924  3.244924
##  [85]  3.169708  3.096236  3.024467  2.954361  2.885880  2.818987  2.753644
##  [92]  2.689816  2.627468  2.566564  2.507073  2.448960  2.392194  2.336744
##  [99]  2.282580  2.229671  2.177988
## 
## 
## [[37]]
## [[37]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[37]]$y
##   [1]  0.000000  4.209304  7.437149  9.891657 11.737255 13.103839 14.094013
##   [8] 14.788817 15.252247 15.534831 15.676454 15.708594 15.656080 15.538493
##  [15] 15.371263 15.166545 14.933903 14.680855 14.413304 14.135877 13.852193
##  [22] 13.565078 13.276727 12.988842 12.702734 12.419404 12.139614 11.863933
##  [29] 11.592780 11.326455 11.065169 10.809056 10.558196 10.312626 10.072347
##  [36]  9.837334  9.607541  9.382908  9.163363  8.948826  8.739209  8.534421
##  [43]  8.334367  8.138953  7.948081  7.761653  7.579573  7.401745  7.228073
##  [50]  7.058464  6.892825  6.731065  6.573096  6.418829  6.268179  6.121061
##  [57]  5.977394  5.837098  5.700092  5.566301  5.435650  5.308064  5.183473
##  [64]  5.061806  4.942994  4.826970  4.713670  4.603029  4.494984  4.389476
##  [71]  4.286444  4.185831  4.087579  3.991633  3.897939  3.806445  3.717098
##  [78]  3.629848  3.544646  3.461444  3.380196  3.300854  3.223374  3.147713
##  [85]  3.073829  3.001678  2.931221  2.862418  2.795229  2.729618  2.665547
##  [92]  2.602980  2.541881  2.482217  2.423953  2.367056  2.311496  2.257239
##  [99]  2.204256  2.152516  2.101991
## 
## 
## [[38]]
## [[38]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[38]]$y
##   [1]  0.000000  4.346692  7.643785 10.122556 11.963788 13.308783 14.267976
##   [8] 14.927667 15.355284 15.603501 15.713456 15.717270 15.640012 15.501238
##  [15] 15.316198 15.096770 14.852203 14.589686 14.314800 14.031873 13.744246
##  [22] 13.454500 13.164611 12.876094 12.590094 12.307477 12.028884 11.754784
##  [29] 11.485515 11.221307 10.962313 10.708620 10.460269 10.217266  9.979585
##  [36]  9.747180  9.519991  9.297942  9.080952  8.868932  8.661788  8.459424
##  [43]  8.261743  8.068647  7.880036  7.695812  7.515879  7.340139  7.168498
##  [50]  7.000863  6.837141  6.677243  6.521081  6.368569  6.219620  6.074154
##  [57]  5.932088  5.793344  5.657844  5.525513  5.396276  5.270061  5.146798
##  [64]  5.026418  4.908853  4.794038  4.681909  4.572401  4.465456  4.361011
##  [71]  4.259009  4.159393  4.062107  3.967096  3.874308  3.783690  3.695191
##  [78]  3.608763  3.524355  3.441923  3.361418  3.282796  3.206013  3.131026
##  [85]  3.057792  2.986272  2.916425  2.848211  2.781593  2.716533  2.652994
##  [92]  2.590942  2.530341  2.471158  2.413359  2.356911  2.301784  2.247947
##  [99]  2.195368  2.144020  2.093872
## 
## 
## [[39]]
## [[39]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[39]]$y
##   [1]  0.000000  4.520976  7.913945 10.436679 12.288541 13.623710 14.561407
##   [8] 15.193812 15.592188 15.811615 15.894656 15.874191 15.775611 15.618512
##  [15] 15.418013 15.185763 14.930735 14.659828 14.378339 14.090327 13.798893
##  [22] 13.506399 13.214636 12.924952 12.638356 12.355595 12.077212 11.803595
##  [29] 11.535013 11.271642 11.013589 10.760905 10.513602 10.271660 10.035036
##  [36]  9.803671  9.577491  9.356414  9.140351  8.929209  8.722890  8.521297
##  [43]  8.324331  8.131894  7.943886  7.760209  7.580769  7.405468  7.234215
##  [50]  7.066916  6.903482  6.743825  6.587858  6.435496  6.286656  6.141258
##  [57]  5.999221  5.860469  5.724925  5.592516  5.463169  5.336814  5.213380
##  [64]  5.092802  4.975012  4.859946  4.747542  4.637737  4.530472  4.425687
##  [71]  4.323327  4.223333  4.125653  4.030231  3.937017  3.845959  3.757006
##  [78]  3.670111  3.585226  3.502304  3.421300  3.342169  3.264869  3.189357
##  [85]  3.115591  3.043531  2.973137  2.904372  2.837198  2.771577  2.707474
##  [92]  2.644853  2.583681  2.523923  2.465548  2.408523  2.352817  2.298399
##  [99]  2.245239  2.193310  2.142581
## 
## 
## [[40]]
## [[40]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[40]]$y
##   [1]  0.000000  4.049549  7.143676  9.487576 11.242804 12.536543 13.468910
##   [8] 14.118704 14.547939 14.805408 14.929492 14.950375 14.891787 14.772374
##  [15] 14.606778 14.406491 14.180526 13.935941 13.678256 13.411785 13.139886
##  [22] 12.865171 12.589661 12.314916 12.042132 11.772216 11.505855 11.243558
##  [29] 10.985696 10.732531 10.484244 10.240948 10.002704  9.769536  9.541436
##  [36]  9.318371  9.100294  8.887141  8.678839  8.475308  8.276463  8.082213
##  [43]  7.892469  7.707136  7.526121  7.349332  7.176674  7.008056  6.843387
##  [50]  6.682577  6.525537  6.372182  6.222425  6.076185  5.933378  5.793925
##  [57]  5.657747  5.524769  5.394915  5.268112  5.144289  5.023376  4.905304
##  [64]  4.790007  4.677420  4.567479  4.460122  4.355288  4.252918  4.152954
##  [71]  4.055340  3.960020  3.866940  3.776049  3.687293  3.600624  3.515992
##  [78]  3.433349  3.352649  3.273845  3.196894  3.121752  3.048375  2.976724
##  [85]  2.906756  2.838433  2.771717  2.706568  2.642950  2.580828  2.520166
##  [92]  2.460930  2.403086  2.346602  2.291446  2.237585  2.184991  2.133633
##  [99]  2.083483  2.034511  1.986690
## 
## 
## [[41]]
## [[41]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[41]]$y
##   [1]  0.000000  4.353689  7.677098 10.192124 12.073341 13.458076 14.454353
##   [8] 15.147128 15.603209 15.875121 16.004152 16.022746 15.956387 15.825085
##  [15] 15.644544 15.427075 15.182327 14.917849 14.639543 14.352012 14.058840
##  [22] 13.762808 13.466070 13.170282 12.876713 12.586327 12.299848 12.017814
##  [29] 11.740613 11.468523 11.201728 10.940345 10.684437 10.434023 10.189091
##  [36]  9.949604  9.715507  9.486729  9.263190  9.044802  8.831470  8.623098
##  [43]  8.419586  8.220832  8.026734  7.837191  7.652103  7.471368  7.294889
##  [50]  7.122568  6.954308  6.790018  6.629603  6.472974  6.320042  6.170721
##  [57]  6.024926  5.882574  5.743584  5.607877  5.475376  5.346005  5.219690
##  [64]  5.096359  4.975942  4.858370  4.743576  4.631494  4.522061  4.415213
##  [71]  4.310889  4.209031  4.109579  4.012477  3.917669  3.825101  3.734721
##  [78]  3.646476  3.560316  3.476192  3.394056  3.313860  3.235559  3.159109
##  [85]  3.084464  3.011584  2.940425  2.870948  2.803113  2.736880  2.672212
##  [92]  2.609073  2.547425  2.487234  2.428465  2.371084  2.315060  2.260359
##  [99]  2.206950  2.154804  2.103890
## 
## 
## [[42]]
## [[42]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[42]]$y
##   [1]  0.000000  4.352992  7.673114 10.183711 12.060287 13.440748 14.433441
##   [8] 15.123483 15.577729 15.848684 15.977570 15.996745 15.931597 15.802039
##  [15] 15.623678 15.408742 15.166799 14.905331 14.630179 14.345894 14.056018
##  [22] 13.763297 13.469854 13.177320 12.886946 12.599679 12.316232 12.037133
##  [29] 11.762765 11.493400 11.229220 10.970339 10.716820 10.468683 10.225916
##  [36]  9.988484  9.756333  9.529396  9.307594  9.090843  8.879051  8.672123
##  [43]  8.469964  8.272474  8.079555  7.891109  7.707037  7.527243  7.351629
##  [50]  7.180103  7.012571  6.848942  6.689126  6.533035  6.380584  6.231688
##  [57]  6.086264  5.944233  5.805515  5.670033  5.537712  5.408479  5.282261
##  [64]  5.158988  5.038592  4.921005  4.806162  4.694000  4.584455  4.477466
##  [71]  4.372974  4.270920  4.171248  4.073902  3.978828  3.885973  3.795284
##  [78]  3.706712  3.620207  3.535721  3.453207  3.372618  3.293910  3.217039
##  [85]  3.141962  3.068636  2.997022  2.927080  2.858769  2.792053  2.726894
##  [92]  2.663255  2.601102  2.540399  2.481113  2.423210  2.366659  2.311427
##  [99]  2.257484  2.204801  2.153346
## 
## 
## [[43]]
## [[43]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[43]]$y
##   [1]  0.000000  3.878498  6.889016  9.206968 10.972728 12.298618 13.274504
##   [8] 13.972277 14.449437 14.751969 14.916636 14.972822 14.944005 14.848936
##  [15] 14.702580 14.516878 14.301345 14.063558 13.809546 13.544093 13.270993
##  [22] 12.993246 12.713219 12.432771 12.153354 11.876101 11.601883 11.331369
##  [29] 11.065059 10.803327 10.546442 10.294590 10.047891  9.806416  9.570192
##  [36]  9.339218  9.113466  8.892888  8.677422  8.466997  8.261529  8.060931
##  [43]  7.865111  7.673974  7.487422  7.305357  7.127682  6.954297  6.785106
##  [50]  6.620011  6.458918  6.301732  6.148362  5.998716  5.852706  5.710245
##  [57]  5.571247  5.435630  5.303311  5.174211  5.048252  4.925358  4.805454
##  [64]  4.688469  4.574331  4.462971  4.354321  4.248317  4.144892  4.043985
##  [71]  3.945535  3.849481  3.755766  3.664332  3.575124  3.488087  3.403169
##  [78]  3.320319  3.239486  3.160620  3.083675  3.008602  2.935358  2.863896
##  [85]  2.794174  2.726150  2.659781  2.595029  2.531853  2.470214  2.410077
##  [92]  2.351403  2.294158  2.238307  2.183815  2.130650  2.078779  2.028171
##  [99]  1.978795  1.930621  1.883619
## 
## 
## [[44]]
## [[44]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[44]]$y
##   [1]  0.000000  4.274092  7.554340 10.052061 11.934032 13.331875 14.349451
##   [8] 15.068715 15.554324 15.857279 16.017805 16.067620 16.031723 15.929814
##  [15] 15.777410 15.586723 15.367360 15.126872 14.871186 14.604949 14.331797
##  [22] 14.054571 13.775483 13.496247 13.218191 12.942331 12.669444 12.400116
##  [29] 12.134782 11.873759 11.617273 11.365477 11.118467 10.876296 10.638982
##  [36] 10.406516 10.178869  9.955996  9.737841  9.524338  9.315415  9.110995
##  [43]  8.910998  8.715341  8.523941  8.336713  8.153574  7.974438  7.799223
##  [50]  7.627845  7.460224  7.296278  7.135930  6.979101  6.825715  6.675697
##  [57]  6.528974  6.385473  6.245126  6.107862  5.973614  5.842316  5.713903
##  [64]  5.588313  5.465482  5.345352  5.227861  5.112952  5.000570  4.890657
##  [71]  4.783160  4.678026  4.575202  4.474639  4.376286  4.280095  4.186018
##  [78]  4.094008  4.004022  3.916013  3.829938  3.745756  3.663423  3.582901
##  [85]  3.504148  3.427127  3.351798  3.278125  3.206071  3.135601  3.066680
##  [92]  2.999274  2.933350  2.868874  2.805816  2.744144  2.683827  2.624836
##  [99]  2.567142  2.510716  2.455530
## 
## 
## [[45]]
## [[45]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[45]]$y
##   [1]  0.000000  4.628162  8.165012 10.845261 12.853609 14.335389 15.404950
##   [8] 16.152251 16.648053 16.948006 17.095868 17.126036 17.065539 16.935609
##  [15] 16.752914 16.530529 16.278708 16.005477 15.717116 15.418528 15.113534
##  [22] 14.805103 14.495536 14.186609 13.879684 13.575798 13.275735 12.980078
##  [29] 12.689256 12.403572 12.123237 11.848385 11.579095 11.315397 11.057288
##  [36] 10.804741 10.557704 10.316114 10.079892  9.848953  9.623207  9.402558
##  [43]  9.186907  8.976153  8.770196  8.568936  8.372270  8.180100  7.992325
##  [50]  7.808850  7.629578  7.454414  7.283266  7.116044  6.952657  6.793019
##  [57]  6.637044  6.484649  6.335751  6.190272  6.048132  5.909255  5.773567
##  [64]  5.640994  5.511464  5.384909  5.261260  5.140450  5.022413  4.907087
##  [71]  4.794409  4.684318  4.576756  4.471663  4.368983  4.268661  4.170642
##  [78]  4.074874  3.981306  3.889886  3.800565  3.713295  3.628029  3.544721
##  [85]  3.463325  3.383799  3.306099  3.230183  3.156011  3.083541  3.012736
##  [92]  2.943556  2.875965  2.809926  2.745404  2.682363  2.620770  2.560590
##  [99]  2.501793  2.444346  2.388218
## 
## 
## [[46]]
## [[46]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[46]]$y
##   [1]  0.000000  4.002791  7.111841  9.506770 11.331543 12.701538 13.709210
##   [8] 14.428635 14.919155 15.228297 15.394116 15.447074 15.411545 15.307021
##  [15] 15.149086 14.950185 14.720252 14.467206 14.197353 13.915706 13.626241
##  [22] 13.332108 13.035789 12.739238 12.443983 12.151210 11.861836 11.576560
##  [29] 11.295908 11.020269 10.749920 10.485053 10.225791  9.972201  9.724308
##  [36]  9.482103  9.245549  9.014592  8.789161  8.569170  8.354529  8.145138
##  [43]  7.940894  7.741690  7.547419  7.357971  7.173236  6.993106  6.817473
##  [50]  6.646229  6.479269  6.316489  6.157788  6.003066  5.852224  5.705166
##  [57]  5.561799  5.422031  5.285773  5.152936  5.023436  4.897189  4.774114
##  [64]  4.654130  4.537161  4.423132  4.311967  4.203597  4.097949  3.994956
##  [71]  3.894552  3.796671  3.701250  3.608226  3.517541  3.429135  3.342951
##  [78]  3.258932  3.177026  3.097178  3.019336  2.943451  2.869474  2.797355
##  [85]  2.727049  2.658510  2.591694  2.526557  2.463057  2.401153  2.340804
##  [92]  2.281973  2.224620  2.168709  2.114202  2.061066  2.009265  1.958766
##  [99]  1.909537  1.861544  1.814758
## 
## 
## [[47]]
## [[47]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[47]]$y
##   [1]  0.000000  3.675169  6.509360  8.677989 10.320227 11.546482 12.444337
##   [8] 13.083259 13.518340 13.793265 13.942667 13.993995 13.969001 13.884910
##  [15] 13.755361 13.591146 13.400798 13.191056 12.967241 12.733544 12.493264
##  [22] 12.248988 12.002744 11.756114 11.510327 11.266333 11.024858 10.786458
##  [29] 10.551546 10.320429 10.093327  9.870390  9.651719  9.437369  9.227364
##  [36]  9.021704  8.820366  8.623314  8.430499  8.241866  8.057349  7.876881
##  [43]  7.700390  7.527802  7.359041  7.194031  7.032694  6.874956  6.720739
##  [50]  6.569968  6.422570  6.278470  6.137597  5.999880  5.865249  5.733635
##  [57]  5.604973  5.479195  5.356238  5.236040  5.118537  5.003671  4.891382
##  [64]  4.781612  4.674305  4.569406  4.466861  4.366617  4.268622  4.172827
##  [71]  4.079181  3.987637  3.898147  3.810665  3.725147  3.641547  3.559824
##  [78]  3.479935  3.401839  3.325495  3.250865  3.177909  3.106591  3.036873
##  [85]  2.968720  2.902096  2.836967  2.773300  2.711062  2.650221  2.590745
##  [92]  2.532603  2.475767  2.420206  2.365892  2.312797  2.260893  2.210154
##  [99]  2.160554  2.112067  2.064668
## 
## 
## [[48]]
## [[48]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[48]]$y
##   [1]  0.000000  4.320966  7.668596 10.241477 12.198114 13.665016 14.743141
##   [8] 15.513047 16.039008 16.372290 16.553782 16.616081 16.585168 16.481742
##  [15] 16.322285 16.119913 15.885055 15.625999 15.349317 15.060219 14.762826
##  [22] 14.460388 14.155465 13.850063 13.545748 13.243737 12.944965 12.650148
##  [29] 12.359824 12.074391 11.794136 11.519258 11.249885 10.986094 10.727914
##  [36] 10.475344 10.228355  9.986897  9.750907  9.520307  9.295012  9.074930
##  [43]  8.859964  8.650015  8.444981  8.244758  8.049244  7.858336  7.671931
##  [50]  7.489928  7.312227  7.138730  6.969339  6.803960  6.642499  6.484864
##  [57]  6.330966  6.180717  6.034031  5.890825  5.751015  5.614523  5.481269
##  [64]  5.351176  5.224171  5.100179  4.979130  4.860953  4.745582  4.632948
##  [71]  4.522987  4.415636  4.310833  4.208517  4.108629  4.011113  3.915910
##  [78]  3.822968  3.732231  3.643648  3.557167  3.472739  3.390315  3.309847
##  [85]  3.231288  3.154595  3.079722  3.006625  2.935264  2.865597  2.797583
##  [92]  2.731183  2.666359  2.603074  2.541291  2.480974  2.422089  2.364601
##  [99]  2.308478  2.253687  2.200197
## 
## 
## [[49]]
## [[49]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[49]]$y
##   [1]  0.000000  4.277458  7.539257 10.004447 11.845317 13.197365 14.167142
##   [8] 14.838426 15.277081 15.534873 15.652478 15.661851 15.588083 15.450866
##  [15] 15.265649 15.044537 14.797011 14.530486 14.250752 13.962325 13.668714
##  [22] 13.372640 13.076208 12.781031 12.488346 12.199085 11.913949 11.633454
##  [29] 11.357972 11.087762 10.822997 10.563781 10.310165 10.062159  9.819742
##  [36]  9.582870  9.351478  9.125490  8.904820  8.689372  8.479048  8.273744
##  [43]  8.073356  7.877778  7.686904  7.500627  7.318844  7.141449  6.968341
##  [50]  6.799419  6.634583  6.473738  6.316786  6.163636  6.014196  5.868377
##  [57]  5.726091  5.587254  5.451782  5.319593  5.190609  5.064752  4.941946
##  [64]  4.822118  4.705195  4.591106  4.479784  4.371161  4.265171  4.161752
##  [71]  4.060840  3.962375  3.866297  3.772549  3.681074  3.591817  3.504724
##  [78]  3.419743  3.336823  3.255913  3.176966  3.099932  3.024766  2.951423
##  [85]  2.879858  2.810029  2.741893  2.675409  2.610536  2.547237  2.485473
##  [92]  2.425207  2.366401  2.309022  2.253034  2.198403  2.145097  2.093084
##  [99]  2.042332  1.992810  1.944490
## 
## 
## [[50]]
## [[50]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[50]]$y
##   [1]  0.000000  4.165401  7.404528  9.904516 11.815077 13.255975 14.323008
##   [8] 15.092795 15.626608 15.973441 16.172457 16.254958 16.245953 16.165418
##  [15] 16.029300 15.850324 15.638635 15.402318 15.147807 14.880217 14.603608
##  [22] 14.321195 14.035521 13.748589 13.461970 13.176893 12.894311 12.614960
##  [29] 12.339399 12.068049 11.801220 11.539135 11.281945 11.029748 10.782596
##  [36] 10.540509 10.303479 10.071475  9.844454  9.622355  9.405112  9.192649
##  [43]  8.984886  8.781740  8.583122  8.388946  8.199122  8.013561  7.832173
##  [50]  7.654870  7.481564  7.312169  7.146598  6.984767  6.826594  6.671998
##  [57]  6.520898  6.373217  6.228877  6.087804  5.949925  5.815166  5.683459
##  [64]  5.554734  5.428923  5.305962  5.185785  5.068329  4.953534  4.841338
##  [71]  4.731684  4.624513  4.519769  4.417397  4.317344  4.219557  4.123985
##  [78]  4.030578  3.939286  3.850062  3.762859  3.677631  3.594333  3.512922
##  [85]  3.433355  3.355590  3.279587  3.205305  3.132705  3.061750  2.992402
##  [92]  2.924624  2.858382  2.793640  2.730365  2.668523  2.608081  2.549008
##  [99]  2.491274  2.434847  2.379698
## 
## 
## [[51]]
## [[51]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[51]]$y
##   [1]  0.000000  4.391281  7.761627 10.325686 12.253482 13.679679 14.710926
##   [8] 15.431669 15.908764 16.195129 16.332641 16.354426 16.286677 16.150098
##  [15] 15.961042 15.732416 15.474397 15.195001 14.900532 14.595940 14.285101
##  [22] 13.971043 13.656122 13.342162 13.030570 12.722417 12.418517 12.119474
##  [29] 11.825730 11.537600 11.255298 10.978960 10.708658 10.444418 10.186230
##  [36]  9.934052  9.687822  9.447463  9.212882  8.983979  8.760647  8.542776
##  [43]  8.330249  8.122952  7.920768  7.723580  7.531272  7.343730  7.160840
##  [50]  6.982490  6.808571  6.638975  6.473597  6.312332  6.155080  6.001742
##  [57]  5.852222  5.706424  5.564256  5.425629  5.290455  5.158648  5.030123
##  [64]  4.904801  4.782600  4.663443  4.547255  4.433962  4.323491  4.215772
##  [71]  4.110737  4.008319  3.908452  3.811073  3.716121  3.623534  3.533254
##  [78]  3.445224  3.359386  3.275688  3.194074  3.114494  3.036897  2.961233
##  [85]  2.887454  2.815513  2.745365  2.676965  2.610268  2.545234  2.481819
##  [92]  2.419985  2.359691  2.300900  2.243573  2.187675  2.133169  2.080021
##  [99]  2.028198  1.977665  1.928392
## 
## 
## [[52]]
## [[52]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[52]]$y
##   [1]  0.000000  4.340735  7.624829 10.084949 11.903102 13.221647 14.151921
##   [8] 14.780983 15.176895 15.392855 15.470428 15.442081 15.333158 15.163434
##  [15] 14.948325 14.699838 14.427315 14.138012 13.837555 13.530294 13.219585
##  [22] 12.908004 12.597521 12.289630 11.985456 11.685834 11.391377 11.102522
##  [29] 10.819570 10.542717 10.272078 10.007707  9.749608  9.497749  9.252070
##  [36]  9.012491  8.778917  8.551240  8.329346  8.113114  7.902420  7.697140
##  [43]  7.497147  7.302314  7.112516  6.927629  6.747532  6.572102  6.401224
##  [50]  6.234780  6.072657  5.914745  5.760936  5.611123  5.465203  5.323077
##  [57]  5.184645  5.049812  4.918484  4.790571  4.665984  4.544637  4.426446
##  [64]  4.311327  4.199203  4.089994  3.983626  3.880023  3.779115  3.680831
##  [71]  3.585104  3.491866  3.401052  3.312601  3.226449  3.142539  3.060810
##  [78]  2.981207  2.903675  2.828158  2.754606  2.682966  2.613190  2.545229
##  [85]  2.479034  2.414562  2.351766  2.290603  2.231031  2.173008  2.116495
##  [92]  2.061451  2.007838  1.955620  1.904760  1.855222  1.806973  1.759979
##  [99]  1.714207  1.669625  1.626203
## 
## 
## [[53]]
## [[53]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[53]]$y
##   [1]  0.000000  4.517361  7.919529 10.456765 12.323731 13.671828 14.618798
##   [8] 15.256193 15.655187 15.871100 15.946916 15.916024 15.804346 15.631998
##  [15] 15.414576 15.164162 14.890107 14.599636 14.298320 13.990447 13.679306
##  [22] 13.367411 13.056673 12.748536 12.444080 12.144106 11.849193 11.559757
##  [29] 11.276080 10.998345 10.726658 10.461065 10.201569  9.948137  9.700710
##  [36]  9.459211  9.223549  8.993621  8.769319  8.550527  8.337131  8.129009
##  [43]  7.926045  7.728117  7.535109  7.346902  7.163382  6.984435  6.809950
##  [50]  6.639817  6.473929  6.312181  6.154472  6.000700  5.850769  5.704582
##  [57]  5.562047  5.423072  5.287569  5.155451  5.026634  4.901035  4.778574
##  [64]  4.659173  4.542756  4.429247  4.318574  4.210666  4.105455  4.002872
##  [71]  3.902853  3.805333  3.710249  3.617542  3.527151  3.439018  3.353088
##  [78]  3.269304  3.187614  3.107966  3.030307  2.954589  2.880763  2.808782
##  [85]  2.738599  2.670170  2.603451  2.538398  2.474972  2.413130  2.352833
##  [92]  2.294043  2.236722  2.180833  2.126341  2.073210  2.021407  1.970898
##  [99]  1.921652  1.873635  1.826819
## 
## 
## [[54]]
## [[54]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[54]]$y
##   [1]  0.000000  4.468487  7.876069 10.449696 12.368355 13.773189 14.775488
##   [8] 15.462983 15.904823 16.155497 16.257925 16.245906 16.146040 15.979251
##  [15] 15.761985 15.507156 15.224895 14.923136 14.608084 14.284577 13.956381
##  [22] 13.626413 13.296926 12.969646 12.645887 12.326640 12.012638 11.704418
##  [29] 11.402359 11.106718 10.817655 10.535260 10.259560  9.990542  9.728158
##  [36]  9.472334  9.222975  8.979974  8.743213  8.512566  8.287901  8.069087
##  [43]  7.855986  7.648463  7.446383  7.249611  7.058015  6.871463  6.689826
##  [50]  6.512979  6.340797  6.173159  6.009948  5.851047  5.696344  5.545728
##  [57]  5.399093  5.256332  5.117346  4.982033  4.850297  4.722044  4.597181
##  [64]  4.475620  4.357273  4.242055  4.129884  4.020678  3.914360  3.810853
##  [71]  3.710083  3.611978  3.516467  3.423481  3.332955  3.244822  3.159019
##  [78]  3.075485  2.994160  2.914986  2.837905  2.762863  2.689805  2.618678
##  [85]  2.549433  2.482018  2.416386  2.352490  2.290283  2.229721  2.170761
##  [92]  2.113360  2.057476  2.003070  1.950103  1.898537  1.848334  1.799459
##  [99]  1.751876  1.705551  1.660451
## 
## 
## [[55]]
## [[55]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[55]]$y
##   [1]  0.000000  3.790205  6.674989  8.851424 10.474098 11.664261 12.517005
##   [8] 13.106901 13.492414 13.719378 13.823713 13.833563 13.770969 13.653187
##  [15] 13.493719 13.303119 13.089633 12.859695 12.618317 12.369398 12.115963
##  [22] 11.860353 11.604373 11.349407 11.096510 10.846482 10.599919 10.357261
##  [29] 10.118825  9.884834  9.655433  9.430714  9.210721  8.995464  8.784928
##  [36]  8.579076  8.377856  8.181206  7.989054  7.801324  7.617932  7.438794
##  [43]  7.263825  7.092936  6.926040  6.763049  6.603878  6.448439  6.296648
##  [50]  6.148423  6.003680  5.862339  5.724322  5.589552  5.457952  5.329448
##  [57]  5.203969  5.081442  4.961800  4.844974  4.730898  4.619507  4.510739
##  [64]  4.404531  4.300825  4.199559  4.100678  4.004125  3.909846  3.817786
##  [71]  3.727893  3.640118  3.554409  3.470718  3.388997  3.309201  3.231283
##  [78]  3.155201  3.080909  3.008367  2.937533  2.868367  2.800829  2.734881
##  [85]  2.670487  2.607608  2.546210  2.486258  2.427717  2.370555  2.314738
##  [92]  2.260236  2.207017  2.155051  2.104309  2.054762  2.006381  1.959139
##  [99]  1.913010  1.867967  1.823984
## 
## 
## [[56]]
## [[56]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[56]]$y
##   [1]  0.000000  4.316860  7.590930 10.051703 11.878648 13.212101 14.161790
##   [8] 14.813497 15.234273 15.476520 15.581180 15.580232 15.498643 15.355899
##  [15] 15.167194 14.944373 14.696656 14.431214 14.153613 13.868169 13.578218
##  [22] 13.286329 12.994475 12.704163 12.416535 12.132450 11.852545 11.577284
##  [29] 11.306999 11.041915 10.782180 10.527877 10.279042 10.035675  9.797745
##  [36]  9.565203  9.337982  9.116003  8.899180  8.687419  8.480622  8.278691
##  [43]  8.081522  7.889013  7.701063  7.517568  7.338429  7.163545  6.992818
##  [50]  6.826152  6.663452  6.504625  6.349580  6.198228  6.050480  5.906253
##  [57]  5.765462  5.628027  5.493866  5.362903  5.235061  5.110267  4.988447
##  [64]  4.869530  4.753449  4.640134  4.529520  4.421543  4.316141  4.213250
##  [71]  4.112813  4.014769  3.919063  3.825638  3.734441  3.645417  3.558515
##  [78]  3.473686  3.390878  3.310044  3.231138  3.154112  3.078922  3.005525
##  [85]  2.933878  2.863939  2.795666  2.729022  2.663966  2.600461  2.538469
##  [92]  2.477956  2.418885  2.361222  2.304934  2.249988  2.196351  2.143994
##  [99]  2.092884  2.042992  1.994290
## 
## 
## [[57]]
## [[57]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[57]]$y
##   [1]  0.000000  4.046480  7.136491  9.476005 11.227061 12.517128 13.446468
##   [8] 14.093935 14.521537 14.778027 14.901725 14.922745 14.864742 14.746290
##  [15] 14.581967 14.383202 14.158955 13.916236 13.660525 13.396100 13.126291
##  [22] 12.853685 12.580284 12.307630 12.036906 11.769010 11.504619 11.244236
##  [29] 10.988228 10.736854 10.490292 10.248653 10.011998  9.780351  9.553704
##  [36]  9.332026  9.115269  8.903371  8.696261  8.493861  8.296086  8.102848
##  [43]  7.914059  7.729627  7.549460  7.373467  7.201557  7.033638  6.869622
##  [50]  6.709422  6.552949  6.400119  6.250849  6.105056  5.962661  5.823585
##  [57]  5.687750  5.555083  5.425509  5.298956  5.175355  5.054636  4.936733
##  [64]  4.821579  4.709111  4.599267  4.491984  4.387204  4.284868  4.184918
##  [71]  4.087301  3.991960  3.898843  3.807898  3.719074  3.632323  3.547595
##  [78]  3.464843  3.384021  3.305085  3.227990  3.152694  3.079153  3.007328
##  [85]  2.937179  2.868666  2.801751  2.736397  2.672567  2.610226  2.549340
##  [92]  2.489874  2.431794  2.375070  2.319669  2.265560  2.212713  2.161099
##  [99]  2.110689  2.061454  2.013369
## 
## 
## [[58]]
## [[58]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[58]]$y
##   [1]  0.000000  4.145736  7.256971  9.567531 11.258960 12.472201 13.316656
##   [8] 13.877233 14.219815 14.395512 14.443958 14.395875 14.275067 14.099963
##  [15] 13.884820 13.640657 13.375976 13.097331 12.809757 12.517118 12.222365
##  [22] 11.927741 11.634942 11.345243 11.059586 10.778663 10.502968 10.232843
##  [29]  9.968517  9.710126  9.457740  9.211377  8.971013  8.736597  8.508055
##  [36]  8.285295  8.068216  7.856707  7.650651  7.449927  7.254415  7.063991
##  [43]  6.878531  6.697915  6.522022  6.350732  6.183928  6.021496  5.863324
##  [50]  5.709300  5.559319  5.413273  5.271062  5.132585  4.997744  4.866444
##  [57]  4.738593  4.614100  4.492877  4.374838  4.259900  4.147982  4.039004
##  [64]  3.932889  3.829561  3.728948  3.630979  3.535583  3.442694  3.352245
##  [71]  3.264172  3.178413  3.094908  3.013596  2.934420  2.857325  2.782255
##  [78]  2.709157  2.637980  2.568673  2.501187  2.435474  2.371487  2.309182
##  [85]  2.248513  2.189438  2.131916  2.075904  2.021365  1.968258  1.916546
##  [92]  1.866193  1.817163  1.769421  1.722933  1.677667  1.633590  1.590671
##  [99]  1.548880  1.508187  1.468562
## 
## 
## [[59]]
## [[59]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[59]]$y
##   [1]  0.000000  4.356523  7.645050 10.105736 11.925174 13.248329 14.187818
##   [8] 14.831117 15.246164 15.485717 15.590734 15.593005 15.517197 15.382438
##  [15] 15.203558 14.992043 14.756782 14.504649 14.240950 13.969773 13.694264
##  [22] 13.416832 13.139320 12.863125 12.589306 12.318652 12.051750 11.789026
##  [29] 11.530782 11.277227 11.028495 10.784665 10.545771 10.311817 10.082778
##  [36]  9.858614  9.639268  9.424673  9.214756  9.009437  8.808632  8.612256
##  [43]  8.420221  8.232440  8.048825  7.869288  7.693742  7.522102  7.354283
##  [50]  7.190202  7.029777  6.872927  6.719574  6.569641  6.423051  6.279731
##  [57]  6.139608  6.002610  5.868669  5.737716  5.609685  5.484510  5.362128
##  [64]  5.242477  5.125496  5.011124  4.899305  4.789981  4.683097  4.578597
##  [71]  4.476429  4.376541  4.278882  4.183402  4.090053  3.998786  3.909556
##  [78]  3.822317  3.737025  3.653636  3.572108  3.492399  3.414469  3.338278
##  [85]  3.263787  3.190958  3.119754  3.050139  2.982077  2.915535  2.850477
##  [92]  2.786870  2.724683  2.663884  2.604442  2.546325  2.489506  2.433955
##  [99]  2.379643  2.326543  2.274628
## 
## 
## [[60]]
## [[60]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[60]]$y
##   [1]  0.000000  4.350874  7.655821 10.145325 11.999498 13.359081 14.334044
##   [8] 15.010299 15.454960 15.720443 15.847680 15.868628 15.808224 15.685925
##  [15] 15.516899 15.312970 15.083342 14.835174 14.574030 14.304223 14.029094
##  [22] 13.751222 13.472590 13.194720 12.918770 12.645617 12.375917 12.110156
##  [29] 11.848685 11.591753 11.339528 11.092115 10.849573 10.611922 10.379153
##  [36] 10.151239  9.928132  9.709775  9.496101  9.287034  9.082497  8.882407
##  [43]  8.686680  8.495231  8.307973  8.124823  7.945692  7.770499  7.599157
##  [50]  7.431586  7.267704  7.107431  6.950688  6.797399  6.647489  6.500883
##  [57]  6.357508  6.217295  6.080173  5.946074  5.814933  5.686684  5.561262
##  [64]  5.438607  5.318657  5.201352  5.086634  4.974446  4.864733  4.757439
##  [71]  4.652512  4.549899  4.449548  4.351412  4.255439  4.161584  4.069798
##  [78]  3.980037  3.892255  3.806410  3.722457  3.640357  3.560067  3.481548
##  [85]  3.404761  3.329667  3.256230  3.184412  3.114178  3.045494  2.978324
##  [92]  2.912636  2.848396  2.785573  2.724136  2.664054  2.605297  2.547836
##  [99]  2.491642  2.436688  2.382946
## 
## 
## [[61]]
## [[61]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[61]]$y
##   [1]  0.000000  4.158579  7.353456  9.786093 11.616332 12.970982 13.950631
##   [8] 14.635059 15.087530 15.358198 15.486815 15.504876 15.437326 15.303908
##  [15] 15.120242 14.898673 14.648952 14.378766 14.094172 13.799928 13.499768
##  [22] 13.196610 12.892726 12.589878 12.289425 11.992403 11.699597 11.411592
##  [29] 11.128815 10.851568 10.580057 10.314411 10.054697  9.800938  9.553119
##  [36]  9.311197  9.075110  8.844777  8.620106  8.400996  8.187340  7.979025
##  [43]  7.775937  7.577961  7.384979  7.196875  7.013533  6.834839  6.660679
##  [50]  6.490943  6.325521  6.164305  6.007191  5.854076  5.704858  5.559441
##  [57]  5.417728  5.279624  5.145039  5.013884  4.886070  4.761514  4.640133
##  [64]  4.521845  4.406572  4.294237  4.184766  4.078085  3.974124  3.872813
##  [71]  3.774084  3.677872  3.584113  3.492744  3.403704  3.316934  3.232376
##  [78]  3.149973  3.069672  2.991417  2.915157  2.840841  2.768420  2.697845
##  [85]  2.629069  2.562047  2.496733  2.433084  2.371058  2.310613  2.251708
##  [92]  2.194306  2.138367  2.083854  2.030730  1.978961  1.928512  1.879348
##  [99]  1.831438  1.784750  1.739251
## 
## 
## [[62]]
## [[62]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[62]]$y
##   [1]  0.000000  4.100815  7.240566  9.624819 11.415575 12.740491 13.700153
##   [8] 14.373807 14.823880 15.099539 15.239501 15.274246 15.227762 15.118918
##  [15] 14.962553 14.770324 14.551388 14.312925 14.060561 13.798698 13.530772
##  [22] 13.259461 12.986842 12.714523 12.443741 12.175443 11.910344 11.648981
##  [29] 11.391751 11.138938 10.890741 10.647290 10.408663 10.174898  9.945999
##  [36]  9.721948  9.502705  9.288218  9.078425  8.873253  8.672627  8.476464
##  [43]  8.284681  8.097192  7.913911  7.734750  7.559623  7.388444  7.221127
##  [50]  7.057588  6.897745  6.741515  6.588818  6.439576  6.293711  6.151147
##  [57]  6.011811  5.875629  5.742531  5.612447  5.485309  5.361050  5.239606
##  [64]  5.120912  5.004907  4.891530  4.780720  4.672421  4.566575  4.463126
##  [71]  4.362021  4.263207  4.166631  4.072242  3.979992  3.889831  3.801713
##  [78]  3.715591  3.631420  3.549156  3.468755  3.390176  3.313377  3.238317
##  [85]  3.164958  3.093261  3.023188  2.954702  2.887768  2.822350  2.758414
##  [92]  2.695926  2.634854  2.575165  2.516829  2.459814  2.404091  2.349630
##  [99]  2.296402  2.244381  2.193538
## 
## 
## [[63]]
## [[63]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[63]]$y
##   [1]  0.000000  4.646106  8.157601 10.788043 12.734816 14.151560 15.157856
##   [8] 15.846766 16.290705 16.546008 16.656494 16.656236 16.571719 16.423521
##  [15] 16.227622 15.996419 15.739528 15.464391 15.176764 14.881091 14.580792
##  [22] 14.278494 13.976206 13.675455 13.377397 13.082895 12.792591 12.506951
##  [29] 12.226305 11.950879 11.680820 11.416211 11.157088 10.903451 10.655273
##  [36] 10.412504 10.175079  9.942923  9.715951  9.494071  9.277191  9.065211
##  [43]  8.858033  8.655558  8.457686  8.264317  8.075354  7.890700  7.710260
##  [50]  7.533938  7.361642  7.193283  7.028771  6.868018  6.710940  6.557452
##  [57]  6.407474  6.260925  6.117728  5.977804  5.841081  5.707484  5.576943
##  [64]  5.449387  5.324749  5.202961  5.083958  4.967678  4.854056  4.743034
##  [71]  4.634551  4.528549  4.424971  4.323763  4.224869  4.128237  4.033816
##  [78]  3.941554  3.851402  3.763312  3.677237  3.593131  3.510948  3.430645
##  [85]  3.352179  3.275507  3.200590  3.127385  3.055855  2.985961  2.917666
##  [92]  2.850933  2.785726  2.722010  2.659752  2.598918  2.539475  2.481392
##  [99]  2.424637  2.369180  2.314992
## 
## 
## [[64]]
## [[64]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[64]]$y
##   [1]  0.000000  4.727591  8.273899 10.909890 12.844869 14.240504 15.221653
##   [8] 15.884740 16.304217 16.537558 16.629127 16.613154 16.516041 16.358144
##  [15] 16.155149 15.919132 15.659385 15.383044 15.095586 14.801204 14.503098
##  [22] 14.203707 13.904878 13.608003 13.314124 13.024011 12.738229 12.457180
##  [29] 12.181145 11.910309 11.644788 11.384638 11.129878 10.880492 10.636443
##  [36] 10.397672 10.164111  9.935680  9.712293  9.493859  9.280284  9.071472
##  [43]  8.867327  8.667751  8.472648  8.281921  8.095477  7.913221  7.735062
##  [50]  7.560908  7.390671  7.224264  7.061602  6.902600  6.747177  6.595252
##  [57]  6.446747  6.301586  6.159692  6.020993  5.885417  5.752894  5.623354
##  [64]  5.496731  5.372960  5.251975  5.133714  5.018116  4.905122  4.794671
##  [71]  4.686708  4.581175  4.478019  4.377186  4.278623  4.182279  4.088105
##  [78]  3.996052  3.906071  3.818116  3.732142  3.648104  3.565958  3.485662
##  [85]  3.407174  3.330453  3.255460  3.182156  3.110502  3.040461  2.971998
##  [92]  2.905076  2.839661  2.775720  2.713218  2.652123  2.592404  2.534030
##  [99]  2.476970  2.421195  2.366676
## 
## 
## [[65]]
## [[65]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[65]]$y
##   [1]  0.000000  4.059099  7.168357  9.529430 11.301608 12.610705 13.556085
##   [8] 14.216211 14.653035 14.915457 15.042069 15.063320 15.003221 14.880700
##  [15] 14.710670 14.504870 14.272533 14.020914 13.755705 13.481363 13.201373
##  [22] 12.918448 12.634698 12.351752 12.070863 11.792986 11.518842 11.248969
##  [29] 10.983759 10.723492 10.468357 10.218475  9.973913  9.734694  9.500809
##  [36]  9.272226  9.048891  8.830736  8.617684  8.409648  8.206537  8.008253
##  [43]  7.814699  7.625775  7.441380  7.261414  7.085776  6.914367  6.747090
##  [50]  6.583849  6.424547  6.269093  6.117394  5.969361  5.824907  5.683946
##  [57]  5.546394  5.412169  5.281190  5.153381  5.028663  4.906964  4.788208
##  [64]  4.672327  4.559250  4.448909  4.341238  4.236173  4.133650  4.033609
##  [71]  3.935988  3.840731  3.747778  3.657075  3.568568  3.482202  3.397926
##  [78]  3.315690  3.235445  3.157141  3.080733  3.006173  2.933418  2.862424
##  [85]  2.793149  2.725549  2.659586  2.595219  2.532410  2.471121  2.411316
##  [92]  2.352957  2.296012  2.240444  2.186221  2.133311  2.081681  2.031300
##  [99]  1.982139  1.934168  1.887357
## 
## 
## [[66]]
## [[66]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[66]]$y
##   [1]  0.000000  4.589636  8.010833 10.533985 12.367542 13.672180 14.571739
##   [8] 15.161650 15.515438 15.689737 15.728156 15.664258 15.523866 15.326829
##  [15] 15.088397 14.820271 14.531416 14.228691 13.917330 13.601318 13.283674
##  [22] 12.966679 12.652040 12.341027 12.034573 11.733351 11.437837 11.148354
##  [29] 10.865110 10.588224 10.317747 10.053681  9.795989  9.544606  9.299446
##  [36]  9.060406  8.827375  8.600233  8.378855  8.163113  7.952878  7.748020
##  [43]  7.548410  7.353921  7.164427  6.979802  6.799924  6.624675  6.453936
##  [50]  6.287592  6.125533  5.967648  5.813830  5.663975  5.517982  5.375751
##  [57]  5.237185  5.102190  4.970674  4.842548  4.717725  4.596118  4.477646
##  [64]  4.362228  4.249785  4.140240  4.033519  3.929548  3.828258  3.729579
##  [71]  3.633443  3.539785  3.448541  3.359649  3.273049  3.188681  3.106487
##  [78]  3.026412  2.948402  2.872402  2.798361  2.726229  2.655956  2.587494
##  [85]  2.520797  2.455819  2.392517  2.330846  2.270764  2.212232  2.155208
##  [92]  2.099654  2.045532  1.992805  1.941437  1.891393  1.842639  1.795142
##  [99]  1.748870  1.703790  1.659872
## 
## 
## [[67]]
## [[67]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[67]]$y
##   [1]  0.000000  4.554396  7.927980 10.403932 12.197966 13.474396 14.358375
##   [8] 14.945214 15.307496 15.500482 15.566238 15.536777 15.436454 15.283791
##  [15] 15.092864 14.874363 14.636399 14.385122 14.125182 13.860093 13.592501
##  [22] 13.324392 13.057248 12.792173 12.529976 12.271248 12.016412 11.765764
##  [29] 11.519503 11.277756 11.040594 10.808048 10.580116 10.356776 10.137985
##  [36]  9.923692  9.713833  9.508339  9.307137  9.110150  8.917301  8.728509
##  [43]  8.543695  8.362781  8.185686  8.012334  7.842646  7.676548  7.513963
##  [50]  7.354819  7.199044  7.046566  6.897317  6.751228  6.608232  6.468265
##  [57]  6.331262  6.197160  6.065899  5.937417  5.811657  5.688560  5.568071
##  [64]  5.450134  5.334694  5.221700  5.111099  5.002841  4.896875  4.793154
##  [71]  4.691630  4.592257  4.494988  4.399779  4.306587  4.215369  4.126083
##  [78]  4.038689  3.953145  3.869413  3.787455  3.707233  3.628710  3.551850
##  [85]  3.476618  3.402980  3.330901  3.260349  3.191291  3.123696  3.057533
##  [92]  2.992771  2.929381  2.867334  2.806601  2.747154  2.688967  2.632011
##  [99]  2.576263  2.521695  2.468283
## 
## 
## [[68]]
## [[68]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[68]]$y
##   [1]  0.000000  4.104649  7.234022  9.598529 11.363649 12.659517 13.588461
##   [8] 14.230931 14.650170 14.895882 15.007118 15.014555 14.942278 14.809189
##  [15] 14.630111 14.416664 14.177943 13.921063 13.651578 13.373817 13.091146
##  [22] 12.806175 12.520918 12.236924 11.955373 11.677162 11.402957 11.133253
##  [29] 10.868404 10.608658 10.354178 10.105061  9.861356  9.623069  9.390180
##  [36]  9.162644  8.940399  8.723368  8.511468  8.304607  8.102688  7.905611
##  [43]  7.713275  7.525577  7.342413  7.163681  6.989281  6.819109  6.653069
##  [50]  6.491062  6.332992  6.178765  6.028289  5.881474  5.738232  5.598476
##  [57]  5.462122  5.329088  5.199292  5.072657  4.949106  4.828563  4.710956
##  [64]  4.596213  4.484265  4.375042  4.268481  4.164514  4.063080  3.964116
##  [71]  3.867563  3.773361  3.681453  3.591785  3.504300  3.418946  3.335671
##  [78]  3.254424  3.175157  3.097820  3.022366  2.948751  2.876928  2.806855
##  [85]  2.738489  2.671788  2.606711  2.543220  2.481275  2.420838  2.361874
##  [92]  2.304346  2.248219  2.193460  2.140034  2.087909  2.037054  1.987438
##  [99]  1.939030  1.891801  1.845723
## 
## 
## [[69]]
## [[69]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[69]]$y
##   [1]  0.000000  4.380882  7.706419 10.208443 12.068330 13.427977 14.398380
##   [8] 15.066356 15.499796 15.751784 15.863815 15.868316 15.790618 15.650499
##  [15] 15.463393 15.241337 14.993706 14.727797 14.449279 14.162549 13.871009
##  [22] 13.577285 13.283393 12.990875 12.700903 12.414360 12.131903 11.854011
##  [29] 11.581030 11.313197 11.050669 10.793536 10.541842 10.295590 10.054758
##  [36]  9.819298  9.589148  9.364232  9.144467  8.929761  8.720020  8.515145
##  [43]  8.315037  8.119595  7.928718  7.742306  7.560258  7.382478  7.208867
##  [50]  7.039331  6.873775  6.712107  6.554238  6.400079  6.249543  6.102546
##  [57]  5.959005  5.818839  5.681970  5.548319  5.417811  5.290372  5.165931
##  [64]  5.044416  4.925760  4.809894  4.696754  4.586275  4.478395  4.373052
##  [71]  4.270187  4.169742  4.071660  3.975884  3.882362  3.791039  3.701864
##  [78]  3.614787  3.529758  3.446730  3.365654  3.286486  3.209179  3.133691
##  [85]  3.059979  2.988001  2.917716  2.849084  2.782066  2.716625  2.652723
##  [92]  2.590325  2.529394  2.469896  2.411798  2.355067  2.299670  2.245576
##  [99]  2.192754  2.141175  2.090809
## 
## 
## [[70]]
## [[70]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[70]]$y
##   [1]  0.000000  4.083431  7.166280  9.473555 11.180071 12.421640 13.303761
##   [8] 13.908378 14.299133 14.525443 14.625673 14.629598 14.560323 14.435763
##  [15] 14.269809 14.073217 13.854316 13.619543 13.373867 13.121121 12.864250
##  [22] 12.605513 12.346637 12.088933 11.833392 11.580758 11.331581 11.086260
##  [29] 10.845083 10.608245 10.375874 10.148046  9.924795  9.706125  9.492016
##  [36]  9.282430  9.077317  8.876616  8.680259  8.488173  8.300280  8.116503
##  [43]  7.936761  7.760973  7.589058  7.420934  7.256523  7.095745  6.938521
##  [50]  6.784775  6.634432  6.487416  6.343656  6.203079  6.065615  5.931197
##  [57]  5.799756  5.671227  5.545547  5.422650  5.302478  5.184967  5.070061
##  [64]  4.957702  4.847832  4.740397  4.635342  4.532616  4.432167  4.333943
##  [71]  4.237896  4.143978  4.052141  3.962339  3.874528  3.788662  3.704700
##  [78]  3.622598  3.542315  3.463812  3.387049  3.311987  3.238588  3.166816
##  [85]  3.096634  3.028008  2.960903  2.895284  2.831120  2.768378  2.707027
##  [92]  2.647035  2.588372  2.531010  2.474919  2.420071  2.366438  2.313994
##  [99]  2.262713  2.212568  2.163534
## 
## 
## [[71]]
## [[71]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[71]]$y
##   [1]  0.000000  3.861317  6.853157  9.153107 10.902859 12.215466 13.181113
##   [8] 13.871742 14.344727 14.645821 14.811501 14.870840 14.847002 14.758437
##  [15] 14.619831 14.442872 14.236847 14.009137 13.765595 13.510859 13.248599
##  [22] 12.981709 12.712468 12.442663 12.173690 11.906634 11.642331 11.381420
##  [29] 11.124382 10.871576 10.623260 10.379615 10.140759  9.906763  9.677658
##  [36]  9.453445  9.234103  9.019593  8.809859  8.604837  8.404456  8.208636
##  [43]  8.017295  7.830347  7.647705  7.469281  7.294985  7.124730  6.958426
##  [50]  6.795987  6.637327  6.482359  6.331001  6.183170  6.038785  5.897767
##  [57]  5.760039  5.625524  5.494149  5.365839  5.240525  5.118136  4.998605
##  [64]  4.881864  4.767849  4.656497  4.547745  4.441532  4.337800  4.236490
##  [71]  4.137546  4.040913  3.946537  3.854365  3.764345  3.676428  3.590564
##  [78]  3.506705  3.424805  3.344818  3.266699  3.190404  3.115892  3.043119
##  [85]  2.972046  2.902633  2.834841  2.768633  2.703970  2.640818  2.579141
##  [92]  2.518905  2.460075  2.402619  2.346505  2.291702  2.238179  2.185905
##  [99]  2.134853  2.084993  2.036297
## 
## 
## [[72]]
## [[72]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[72]]$y
##   [1]  0.000000  3.901804  6.940509  9.288799 11.085193 12.440798 13.444721
##   [8] 14.168420 14.669189 14.992959 15.176541 15.249433 15.235262 15.152946
##  [15] 15.017625 14.841409 14.633976 14.403056 14.154814 13.894161 13.625002
##  [22] 13.350437 13.072921 12.794390 12.516365 12.240036 11.966327 11.695952
##  [29] 11.429451 11.167231 10.909592 10.656745 10.408835 10.165951  9.928141
##  [36]  9.695418  9.467767  9.245155  9.027531  8.814833  8.606987  8.403915
##  [43]  8.205533  8.011751  7.822480  7.637628  7.457102  7.280809  7.108656
##  [50]  6.940552  6.776406  6.616128  6.459629  6.306823  6.157625  6.011950
##  [57]  5.869717  5.730846  5.595257  5.462873  5.333620  5.207423  5.084211
##  [64]  4.963913  4.846461  4.731787  4.619826  4.510514  4.403787  4.299586
##  [71]  4.197851  4.098522  4.001544  3.906860  3.814416  3.724160  3.636039
##  [78]  3.550004  3.466004  3.383991  3.303920  3.225742  3.149415  3.074894
##  [85]  3.002136  2.931099  2.861744  2.794029  2.727917  2.663369  2.600349
##  [92]  2.538819  2.478746  2.420094  2.362830  2.306921  2.252334  2.199040
##  [99]  2.147006  2.096204  2.046603
## 
## 
## [[73]]
## [[73]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[73]]$y
##   [1]  0.000000  4.400336  7.774684 10.341004 12.271396 13.701734 14.739285
##   [8] 15.468723 15.956873 16.256472 16.409124 16.447643 16.397903 16.280297
##  [15] 16.110890 15.902326 15.664555 15.405394 15.130980 14.846120 14.554576
##  [22] 14.259283 13.962523 13.666065 13.371273 13.079190 12.790609 12.506124
##  [29] 12.226175 11.951079 11.681056 11.416252 11.156756 10.902609 10.653818
##  [36] 10.410361 10.172198  9.939269  9.711505  9.488828  9.271151  9.058385
##  [43]  8.850435  8.647208  8.448606  8.254534  8.064894  7.879590  7.698528
##  [50]  7.521615  7.348756  7.179863  7.014845  6.853614  6.696086  6.542175
##  [57]  6.391800  6.244879  6.101333  5.961086  5.824062  5.690187  5.559388
##  [64]  5.431596  5.306741  5.184755  5.065574  4.949132  4.835366  4.724215
##  [71]  4.615620  4.509520  4.405860  4.304582  4.205632  4.108957  4.014504
##  [78]  3.922222  3.832062  3.743974  3.657911  3.573826  3.491674  3.411410
##  [85]  3.332992  3.256376  3.181521  3.108388  3.036935  2.967124  2.898919
##  [92]  2.832281  2.767175  2.703566  2.641418  2.580700  2.521377  2.463418
##  [99]  2.406791  2.351466  2.297412
## 
## 
## [[74]]
## [[74]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[74]]$y
##   [1]  0.000000  4.495184  7.928657 10.526682 12.467868 13.893191 14.913912
##   [8] 15.617825 16.074191 16.337643 16.451256 16.448986 16.357582 16.198110
##  [15] 15.987145 15.737718 15.460063 15.162207 14.850434 14.529656 14.203698
##  [22] 13.875532 13.547456 13.221238 12.898227 12.579444 12.265651 11.957407
##  [29] 11.655111 11.359039 11.069368 10.786201 10.509579 10.239502  9.975930
##  [36]  9.718799  9.468024  9.223507  8.985135  8.752790  8.526348  8.305681
##  [43]  8.090660  7.881153  7.677030  7.478161  7.284419  7.095676  6.911807
##  [50]  6.732690  6.558205  6.388235  6.222663  6.061377  5.904269  5.751229
##  [57]  5.602154  5.456940  5.315490  5.177705  5.043490  4.912754  4.785406
##  [64]  4.661359  4.540527  4.422827  4.308177  4.196500  4.087717  3.981754
##  [71]  3.878538  3.777997  3.680063  3.584667  3.491744  3.401229  3.313062
##  [78]  3.227179  3.143523  3.062035  2.982660  2.905342  2.830029  2.756668
##  [85]  2.685208  2.615601  2.547799  2.481754  2.417421  2.354755  2.293714
##  [92]  2.234256  2.176338  2.119923  2.064969  2.011440  1.959299  1.908509
##  [99]  1.859036  1.810845  1.763904
## 
## 
## [[75]]
## [[75]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[75]]$y
##   [1]  0.000000  4.478796  7.883507 10.448565 12.357731 13.755038 14.753369
##   [8] 15.441181 15.887781 16.147459 16.262734 16.266894 16.185989 16.040398
##  [15] 15.846052 15.615396 15.358142 15.081861 14.792446 14.494474 14.191494
##  [22] 13.886247 13.580844 13.276898 12.975640 12.677997 12.384658 12.096129
##  [29] 11.812775 11.534845 11.262503 10.995846 10.734918 10.479724 10.230238
##  [36]  9.986409  9.748172  9.515444  9.288138  9.066154  8.849393  8.637749
##  [43]  8.431115  8.229384  8.032449  7.840201  7.652536  7.469347  7.290531
##  [50]  7.115987  6.945615  6.779316  6.616994  6.458555  6.303907  6.152960
##  [57]  6.005625  5.861817  5.721452  5.584447  5.450722  5.320199  5.192801
##  [64]  5.068453  4.947083  4.828619  4.712991  4.600132  4.489976  4.382457
##  [71]  4.277513  4.175082  4.075104  3.977520  3.882273  3.789306  3.698566
##  [78]  3.609999  3.523552  3.439176  3.356820  3.276436  3.197977  3.121397
##  [85]  3.046651  2.973694  2.902485  2.832981  2.765141  2.698926  2.634296
##  [92]  2.571214  2.509643  2.449546  2.390888  2.333635  2.277752  2.223208
##  [99]  2.169971  2.118007  2.067289
## 
## 
## [[76]]
## [[76]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[76]]$y
##   [1]  0.000000  3.988154  7.067179  9.427314 11.219307 12.562607 13.551856
##   [8] 14.262036 14.752544 15.070429 15.252950 15.329608 15.323758 15.253881
##  [15] 15.134597 14.977468 14.791628 14.584293 14.361154 14.126699 13.884459
##  [22] 13.637207 13.387119 13.135896 12.884863 12.635047 12.387242 12.142054
##  [29] 11.899944 11.661254 11.426238 11.195076 10.967891 10.744763 10.525734
##  [36] 10.310823 10.100024  9.893316  9.690665  9.492029  9.297355  9.106587
##  [43]  8.919666  8.736527  8.557105  8.381334  8.209147  8.040476  7.875253
##  [50]  7.713412  7.554887  7.399611  7.247520  7.098549  6.952636  6.809720
##  [57]  6.669738  6.532632  6.398343  6.266812  6.137985  6.011805  5.888218
##  [64]  5.767172  5.648613  5.532492  5.418757  5.307360  5.198253  5.091389
##  [71]  4.986722  4.884206  4.783798  4.685454  4.589132  4.494790  4.402387
##  [78]  4.311883  4.223241  4.136420  4.051385  3.968097  3.886522  3.806624
##  [85]  3.728368  3.651721  3.576650  3.503122  3.431105  3.360569  3.291484
##  [92]  3.223818  3.157543  3.092631  3.029054  2.966783  2.905792  2.846056
##  [99]  2.787547  2.730241  2.674114
## 
## 
## [[77]]
## [[77]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[77]]$y
##   [1]  0.000000  4.080473  7.197580  9.558268 11.325463 12.627425 13.565114
##   [8] 14.217994 14.648594 14.906113 15.029251 15.048441 14.987606 14.865543
##  [15] 14.697012 14.493597 14.264381 14.016477 13.755448 13.485638 13.210432
##  [22] 12.932459 12.653755 12.375888 12.100060 11.827187 11.557956 11.292878
##  [29] 11.032325 10.776560 10.525762 10.280042 10.039462  9.804042  9.573772
##  [36]  9.348620  9.128533  8.913447  8.703287  8.497970  8.297408  8.101510
##  [43]  7.910183  7.723331  7.540859  7.362671  7.188672  7.018769  6.852869
##  [50]  6.690880  6.532712  6.378276  6.227486  6.080257  5.936506  5.796151
##  [57]  5.659113  5.525313  5.394675  5.267125  5.142590  5.020998  4.902282
##  [64]  4.786372  4.673202  4.562708  4.454826  4.349495  4.246654  4.146244
##  [71]  4.048209  3.952492  3.859038  3.767793  3.678706  3.591725  3.506801
##  [78]  3.423885  3.342929  3.263887  3.186715  3.111367  3.037800  2.965973
##  [85]  2.895844  2.827374  2.760522  2.695251  2.631524  2.569303  2.508553
##  [92]  2.449240  2.391329  2.334787  2.279583  2.225683  2.173058  2.121677
##  [99]  2.071512  2.022532  1.974710
## 
## 
## [[78]]
## [[78]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[78]]$y
##   [1]  0.000000  4.427652  7.764744 10.257715 12.097763 13.433223 14.379162
##   [8] 15.024834 15.439459 15.676711 15.778203 15.776184 15.695642 15.555928
##  [15] 15.372019 15.155501 14.915326 14.658406 14.390068 14.114411 13.834582
##  [22] 13.552989 13.271467 12.991410 12.713866 12.439620 12.169248 11.903171
##  [29] 11.641684 11.384989 11.133216 10.886438 10.644686 10.407957 10.176225
##  [36]  9.949445  9.727558  9.510494  9.298176  9.090524  8.887450  8.688867
##  [43]  8.494687  8.304819  8.119173  7.937661  7.760194  7.586684  7.417047
##  [50]  7.251197  7.089050  6.930526  6.775544  6.624025  6.475893  6.331073
##  [57]  6.189490  6.051072  5.915750  5.783453  5.654114  5.527668  5.404049
##  [64]  5.283195  5.165043  5.049534  4.936607  4.826206  4.718274  4.612756
##  [71]  4.509597  4.408745  4.310149  4.213758  4.119522  4.027394  3.937326
##  [78]  3.849273  3.763189  3.679029  3.596752  3.516315  3.437677  3.360798
##  [85]  3.285637  3.212158  3.140322  3.070093  3.001434  2.934310  2.868688
##  [92]  2.804533  2.741813  2.680496  2.620550  2.561944  2.504649  2.448636
##  [99]  2.393875  2.340339  2.288000
## 
## 
## [[79]]
## [[79]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[79]]$y
##   [1]  0.000000  4.360166  7.639515 10.082053 11.877226 13.172111 14.080875
##   [8] 14.692131 15.074642 15.281759 15.354860 15.326030 15.220136 15.056438
##  [15] 14.849846 14.611891 14.351479 14.075483 13.789194 13.496677 13.201046
##  [22] 12.904677 12.609374 12.316499 12.027069 11.741837 11.461348 11.185993
##  [29] 10.916036 10.651649 10.392932 10.139931  9.892646  9.651048  9.415084
##  [36]  9.184683  8.959759  8.740218  8.525960  8.316878  8.112866  7.913812
##  [43]  7.719608  7.530141  7.345304  7.164987  6.989083  6.817489  6.650099
##  [50]  6.486813  6.327532  6.172159  6.020598  5.872756  5.728543  5.587870
##  [57]  5.450651  5.316800  5.186236  5.058877  4.934646  4.813465  4.695259
##  [64]  4.579957  4.467485  4.357776  4.250761  4.146373  4.044549  3.945226
##  [71]  3.848342  3.753836  3.661652  3.571731  3.484019  3.398461  3.315003
##  [78]  3.233595  3.154187  3.076728  3.001172  2.927471  2.855580  2.785454
##  [85]  2.717051  2.650327  2.585242  2.521755  2.459827  2.399420  2.340497
##  [92]  2.283020  2.226955  2.172267  2.118922  2.066887  2.016129  1.966618
##  [99]  1.918323  1.871214  1.825262
## 
## 
## [[80]]
## [[80]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[80]]$y
##   [1]  0.000000  4.905385  8.597663 11.351819 13.381023 14.850511 15.888352
##   [8] 16.593799 17.043762 17.297823 17.402135 17.392435 17.296387 17.135395
##  [15] 16.926008 16.681016 16.410291 16.121448 15.820347 15.511493 15.198338
##  [22] 14.883522 14.569052 14.256448 13.946852 13.641115 13.339862 13.043541
##  [29] 12.752471 12.466864 12.186853 11.912510 11.643863 11.380901 11.123589
##  [36] 10.871870 10.625674 10.384918 10.149513  9.919363  9.694369  9.474429
##  [43]  9.259442  9.049304  8.843912  8.643164  8.446960  8.255199  8.067783
##  [50]  7.884615  7.705601  7.530648  7.359664  7.192559  7.029247  6.869642
##  [57]  6.713660  6.561219  6.412238  6.266640  6.124347  5.985285  5.849381
##  [64]  5.716562  5.586758  5.459903  5.335927  5.214766  5.096357  4.980636
##  [71]  4.867543  4.757017  4.649002  4.543439  4.440273  4.339449  4.240915
##  [78]  4.144618  4.050508  3.958535  3.868650  3.780806  3.694956  3.611057
##  [85]  3.529062  3.448929  3.370615  3.294080  3.219283  3.146183  3.074744
##  [92]  3.004927  2.936696  2.870013  2.804845  2.741156  2.678914  2.618085
##  [99]  2.558637  2.500539  2.443760
## 
## 
## [[81]]
## [[81]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[81]]$y
##   [1]  0.000000  4.203365  7.422912  9.868437 11.705425 13.064396 14.048286
##   [8] 14.738266 15.198337 15.478948 15.619862 15.652411 15.601270 15.485870
##  [15] 15.321499 15.120182 14.891366 14.642468 14.379303 14.106423 13.827382
##  [22] 13.544952 13.261284 12.978045 12.696515 12.417673 12.142262 11.870837
##  [29] 11.603805 11.341461 11.084008 10.831578 10.584251 10.342060 10.105008
##  [36]  9.873072  9.646209  9.424362  9.207460  8.995427  8.788181  8.585632
##  [43]  8.387691  8.194266  8.005264  7.820591  7.640155  7.463864  7.291626
##  [50]  7.123351  6.958951  6.798337  6.641425  6.488130  6.338370  6.192064
##  [57]  6.049132  5.909499  5.773087  5.639823  5.509634  5.382450  5.258201
##  [64]  5.136820  5.018241  4.902399  4.789231  4.678674  4.570670  4.465159
##  [71]  4.362084  4.261388  4.163016  4.066915  3.973033  3.881318  3.791720
##  [78]  3.704190  3.618681  3.535146  3.453539  3.373816  3.295933  3.219848
##  [85]  3.145520  3.072907  3.001971  2.932672  2.864973  2.798836  2.734227
##  [92]  2.671109  2.609448  2.549210  2.490363  2.432874  2.376713  2.321847
##  [99]  2.268249  2.215888  2.164735
## 
## 
## [[82]]
## [[82]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[82]]$y
##   [1]  0.000000  4.151918  7.309031  9.687468 11.456897 12.750505 13.672827
##   [8] 14.305897 14.714082 14.947877 15.046880 15.042136 14.957973 14.813444
##  [15] 14.623465 14.399699 14.151258 13.885253 13.607222 13.321471 13.031337
##  [22] 12.739401 12.447648 12.157597 11.870402 11.586932 11.307831 11.033571
##  [29] 10.764485 10.500803 10.242669  9.990168  9.743330  9.502150  9.266596
##  [36]  9.036609  8.812117  8.593034  8.379267  8.170715  7.967272  7.768831
##  [43]  7.575283  7.386517  7.202424  7.022896  6.847823  6.677099  6.510621
##  [50]  6.348283  6.189987  6.035632  5.885121  5.738360  5.595257  5.455720
##  [57]  5.319661  5.186993  5.057634  4.931499  4.808510  4.688588  4.571656
##  [64]  4.457640  4.346467  4.238067  4.132370  4.029308  3.928818  3.830833
##  [71]  3.735292  3.642134  3.551299  3.462729  3.376369  3.292162  3.210055
##  [78]  3.129996  3.051934  2.975819  2.901602  2.829236  2.758674  2.689873
##  [85]  2.622787  2.557375  2.493594  2.431403  2.370764  2.311637  2.253985
##  [92]  2.197770  2.142958  2.089512  2.037399  1.986587  1.937041  1.888731
##  [99]  1.841626  1.795696  1.750911
## 
## 
## [[83]]
## [[83]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[83]]$y
##   [1]  0.000000  4.026372  7.107842  9.446077 11.200128 12.495429 13.430889
##   [8] 14.084495 14.517726 14.779040 14.906620 14.930549 14.874513 14.757160
##  [15] 14.593160 14.394044 14.168871 13.924750 13.667250 13.400728 13.128587
##  [22] 12.853474 12.577444 12.302084 12.028614 11.757963 11.490834 11.227751
##  [29] 10.969096 10.715142 10.466076 10.222018  9.983034  9.749151  9.520364
##  [36]  9.296644  9.077942  8.864196  8.655335  8.451278  8.251940  8.057230
##  [43]  7.867057  7.681327  7.499946  7.322820  7.149855  6.980958  6.816037
##  [50]  6.655001  6.497760  6.344229  6.194319  6.047948  5.905032  5.765490
##  [57]  5.629244  5.496216  5.366331  5.239513  5.115692  4.994797  4.876758
##  [64]  4.761508  4.648981  4.539114  4.431842  4.327106  4.224845  4.125000
##  [71]  4.027515  3.932334  3.839402  3.748666  3.660075  3.573577  3.489123
##  [78]  3.406665  3.326156  3.247550  3.170801  3.095866  3.022702  2.951267
##  [85]  2.881520  2.813422  2.746933  2.682015  2.618631  2.556746  2.496323
##  [92]  2.437328  2.379727  2.323487  2.268576  2.214964  2.162618  2.111509
##  [99]  2.061608  2.012886  1.965316
## 
## 
## [[84]]
## [[84]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[84]]$y
##   [1]  0.000000  3.677251  6.557273  8.795214 10.516423 11.822161 12.794221
##   [8] 13.498663 13.988829 14.307787 14.490296 14.564403 14.552733 14.473524
##  [15] 14.341476 14.168426 13.963897 13.735547 13.489524 13.230759 12.963197
##  [22] 12.689991 12.413651 12.136171 11.859128 11.583762 11.311043 11.041720
##  [29] 10.776370 10.515428 10.259214 10.007958  9.761818  9.520895  9.285240
##  [36]  9.054871  8.829776  8.609919  8.395248  8.185696  7.981187  7.781634
##  [43]  7.586948  7.397033  7.211791  7.031124  6.854930  6.683108  6.515560
##  [50]  6.352184  6.192882  6.037557  5.886113  5.738456  5.594494  5.454135
##  [57]  5.317292  5.183877  5.053805  4.926994  4.803362  4.682830  4.565321
##  [64]  4.450760  4.339072  4.230185  4.124031  4.020540  3.919645  3.821282
##  [71]  3.725387  3.631898  3.540755  3.451899  3.365273  3.280821  3.198488
##  [78]  3.118221  3.039968  2.963679  2.889305  2.816796  2.746108  2.677193
##  [85]  2.610008  2.544509  2.480654  2.418401  2.357710  2.298542  2.240860
##  [92]  2.184624  2.129800  2.076352  2.024245  1.973446  1.923922  1.875640
##  [99]  1.828570  1.782682  1.737944
## 
## 
## [[85]]
## [[85]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[85]]$y
##   [1]  0.000000  3.943013  6.965145  9.261521 10.986352 12.261506 13.183278
##   [8] 13.827750 14.255023 14.512562 14.637847 14.660463 14.603754 14.486133
##  [15] 14.322111 14.123121 13.898159 13.654297 13.397087 13.130881 12.859079
##  [22] 12.584336 12.308714 12.033807 11.760842 11.490757 11.224260 10.961879
##  [29] 10.704000 10.450900 10.202767  9.959720  9.721828  9.489116  9.261577
##  [36]  9.039180  8.821875  8.609599  8.402275  8.199822  8.002149  7.809165
##  [43]  7.620775  7.436882  7.257388  7.082197  6.911211  6.744334  6.581472
##  [50]  6.422531  6.267419  6.116046  5.968324  5.824164  5.683483  5.546198
##  [57]  5.412226  5.281489  5.153908  5.029408  4.907915  4.789356  4.673660
##  [64]  4.560759  4.450584  4.343071  4.238155  4.135773  4.035865  3.938369
##  [71]  3.843229  3.750387  3.659788  3.571378  3.485103  3.400912  3.318755
##  [78]  3.238583  3.160348  3.084002  3.009501  2.936799  2.865854  2.796623
##  [85]  2.729064  2.663137  2.598803  2.536023  2.474759  2.414975  2.356636
##  [92]  2.299706  2.244151  2.189939  2.137035  2.085410  2.035032  1.985871
##  [99]  1.937898  1.891084  1.845400
## 
## 
## [[86]]
## [[86]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[86]]$y
##   [1]  0.000000  3.988220  7.043336  9.364112 11.107384 12.396894 13.330260
##   [8] 13.984487 14.420310 14.685621 14.818177 14.847739 14.797753 14.686686
##  [15] 14.529073 14.336350 14.117503 13.879590 13.628143 13.367495 13.101032
##  [22] 12.831391 12.560622 12.290311 12.021677 11.755654 11.492946 11.234083
##  [29] 10.979453 10.729333 10.483917 10.243330 10.007645  9.776895  9.551080
##  [36]  9.330176  9.114142  8.902920  8.696444  8.494639  8.297424  8.104713
##  [43]  7.916421  7.732457  7.552732  7.377156  7.205639  7.038092  6.874427
##  [50]  6.714557  6.558396  6.405859  6.256865  6.111333  5.969182  5.830334
##  [57]  5.694714  5.562248  5.432861  5.306483  5.183044  5.062476  4.944711
##  [64]  4.829686  4.717337  4.607600  4.500416  4.395726  4.293470  4.193594
##  [71]  4.096040  4.000756  3.907688  3.816785  3.727997  3.641274  3.556569
##  [78]  3.473834  3.393024  3.314093  3.236999  3.161698  3.088149  3.016310
##  [85]  2.946143  2.877608  2.810668  2.745284  2.681422  2.619045  2.558119
##  [92]  2.498611  2.440487  2.383715  2.328263  2.274102  2.221200  2.169530
##  [99]  2.119061  2.069766  2.021618
## 
## 
## [[87]]
## [[87]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[87]]$y
##   [1]  0.000000  4.351333  7.650731 10.128242 11.964165 13.299793 14.245838
##   [8] 14.889037 15.297328 15.523916 15.610451 15.589528 15.486642 15.321725
##  [15] 15.110346 14.864656 14.594130 14.306140 14.006416 13.699401 13.388527
##  [22] 13.076440 12.765166 12.456247 12.150850 11.849845 11.553873 11.263395
##  [29] 10.978733 10.700100 10.427625 10.161374  9.901359  9.647559  9.399919
##  [36]  9.158366  8.922809  8.693144  8.469260  8.251040  8.038363  7.831105
##  [43]  7.629142  7.432348  7.240601  7.053777  6.871755  6.694415  6.521641
##  [50]  6.353316  6.189329  6.029570  5.873929  5.722303  5.574588  5.430685
##  [57]  5.290494  5.153921  5.020873  4.891258  4.764989  4.641979  4.522144
##  [64]  4.405403  4.291675  4.180882  4.072950  3.967804  3.865373  3.765585
##  [71]  3.668374  3.573672  3.481415  3.391540  3.303985  3.218690  3.135597
##  [78]  3.054649  2.975791  2.898968  2.824129  2.751222  2.680197  2.611006
##  [85]  2.543601  2.477936  2.413966  2.351647  2.290938  2.231795  2.174180
##  [92]  2.118052  2.063372  2.010105  1.958212  1.907660  1.858412  1.810435
##  [99]  1.763698  1.718166  1.673811
## 
## 
## [[88]]
## [[88]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[88]]$y
##   [1]  0.000000  4.356268  7.699148 10.243898 12.160470 13.583029 14.617486
##   [8] 15.347435 15.838850 16.143789 16.303320 16.349836 16.308876 16.200572
##  [15] 16.040783 15.841994 15.614029 15.364609 15.099799 14.824350 14.541983
##  [22] 14.255602 13.967468 13.679336 13.392560 13.108179 12.826985 12.549573
##  [29] 12.276385 12.007743 11.743875 11.484932 11.231010 10.982158 10.738390
##  [36] 10.499694 10.266036 10.037367  9.813624  9.594737  9.380628  9.171214
##  [43]  8.966411  8.766129  8.570281  8.378777  8.191526  8.008440  7.829431
##  [50]  7.654410  7.483292  7.315992  7.152426  6.992512  6.836169  6.683319
##  [57]  6.533885  6.387789  6.244959  6.105321  5.968805  5.835340  5.704860
##  [64]  5.577296  5.452584  5.330661  5.211464  5.094932  4.981005  4.869626
##  [71]  4.760738  4.654284  4.550210  4.448464  4.348992  4.251745  4.156673
##  [78]  4.063726  3.972858  3.884021  3.797171  3.712263  3.629254  3.548101
##  [85]  3.468762  3.391197  3.315367  3.241233  3.168756  3.097900  3.028628
##  [92]  2.960906  2.894697  2.829969  2.766689  2.704823  2.644341  2.585211
##  [99]  2.527404  2.470889  2.415638
## 
## 
## [[89]]
## [[89]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[89]]$y
##   [1]  0.000000  4.475993  7.875349 10.434286 12.337672 13.730184 14.725029
##   [8] 15.410781 15.856732 16.117075 16.234189 16.241202 16.164003 16.022811
##  [15] 15.833412 15.608118 15.356527 15.086110 14.802673 14.510723 14.213749
##  [22] 13.914442 13.614875 13.316631 13.020914 12.728633 12.440465 12.156906
##  [29] 11.878312 11.604931 11.336925 11.074390 10.817374 10.565883 10.319895
##  [36] 10.079364  9.844229  9.614414  9.389835  9.170401  8.956016  8.746580
##  [43]  8.541994  8.342155  8.146962  7.956312  7.770106  7.588243  7.410626
##  [50]  7.237157  7.067743  6.902288  6.740703  6.582897  6.428783  6.278275
##  [57]  6.131289  5.987743  5.847557  5.710652  5.576952  5.446381  5.318868
##  [64]  5.194339  5.072726  4.953959  4.837974  4.724703  4.614085  4.506056
##  [71]  4.400557  4.297527  4.196910  4.098649  4.002688  3.908973  3.817453
##  [78]  3.728076  3.640791  3.555550  3.472304  3.391008  3.311615  3.234080
##  [85]  3.158361  3.084415  3.012200  2.941676  2.872803  2.805542  2.739857
##  [92]  2.675709  2.613063  2.551884  2.492137  2.433789  2.376807  2.321159
##  [99]  2.266814  2.213741  2.161911
## 
## 
## [[90]]
## [[90]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[90]]$y
##   [1]  0.000000  4.003555  7.038745  9.316547 11.002529 12.226614 13.090732
##   [8] 13.674843 14.041656 14.240339 14.309429 14.279117 14.173046 14.009714
##  [15] 13.803590 13.565970 13.305671 13.029558 12.742967 12.450036 12.153962
##  [22] 11.857207 11.561655 11.268738 10.979535 10.694848 10.415264 10.141203
##  [29]  9.872951  9.610696  9.354544  9.104542  8.860690  8.622952  8.391265
##  [36]  8.165546  7.945697  7.731609  7.523165  7.320243  7.122717  6.930462
##  [43]  6.743348  6.561249  6.384039  6.211592  6.043785  5.880497  5.721610
##  [50]  5.567007  5.416575  5.270203  5.127781  4.989206  4.854372  4.723181
##  [57]  4.595533  4.471334  4.350490  4.232912  4.118511  4.007201  3.898899
##  [64]  3.793524  3.690997  3.591240  3.494179  3.399742  3.307857  3.218455
##  [71]  3.131469  3.046835  2.964487  2.884366  2.806409  2.730560  2.656761
##  [78]  2.584956  2.515092  2.447116  2.380977  2.316626  2.254014  2.193095
##  [85]  2.133821  2.076150  2.020038  1.965442  1.912321  1.860637  1.810349
##  [92]  1.761420  1.713814  1.667494  1.622427  1.578577  1.535912  1.494401
##  [99]  1.454012  1.414714  1.376478
## 
## 
## [[91]]
## [[91]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[91]]$y
##   [1]  0.000000  4.657157  8.188945 10.845559 12.821988 14.270192 15.308584
##   [8] 16.029432 16.504619 16.790142 16.929614 16.956994 16.898717 16.775353
##  [15] 16.602902 16.393802 16.157714 15.902136 15.632879 15.354442 15.070301
##  [22] 14.783133 14.494998 14.207468 13.921744 13.638732 13.359109 13.083379
##  [29] 12.811906 12.544948 12.282683 12.025223 11.772630 11.524932 11.282124
##  [36] 11.044181 10.811061 10.582710 10.359064 10.140051  9.925597  9.715623
##  [43]  9.510046  9.308786  9.111758  8.918880  8.730068  8.545241  8.364317
##  [50]  8.187216  8.013859  7.844168  7.678066  7.515479  7.356332  7.200554
##  [57]  7.048073  6.898821  6.752728  6.609728  6.469756  6.332747  6.198640
##  [64]  6.067373  5.938885  5.813118  5.690014  5.569517  5.451572  5.336124
##  [71]  5.223121  5.112511  5.004244  4.898269  4.794539  4.693005  4.593621
##  [78]  4.496343  4.401124  4.307921  4.216692  4.127396  4.039990  3.954435
##  [85]  3.870692  3.788723  3.708489  3.629954  3.553083  3.477839  3.404189
##  [92]  3.332099  3.261535  3.192466  3.124859  3.058684  2.993910  2.930508
##  [99]  2.868449  2.807704  2.748245
## 
## 
## [[92]]
## [[92]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[92]]$y
##   [1]  0.000000  4.375110  7.691552 10.181919 12.028228 13.372905 14.327382
##   [8] 14.978831 15.395434 15.630506 15.725730 15.713677 15.619793 15.463944
##  [15] 15.261630 15.024934 14.763266 14.483945 14.192651 13.893788 13.590757
##  [22] 13.286176 12.982055 12.679923 12.380937 12.085966 11.795648 11.510447
##  [29] 11.230690 10.956596 10.688302 10.425883 10.169362  9.918727  9.673936
##  [36]  9.434925  9.201615  8.973914  8.751723  8.534935  8.323441  8.117127
##  [43]  7.915880  7.719586  7.528131  7.341401  7.159286  6.981674  6.808458
##  [50]  6.639531  6.474788  6.314128  6.157451  6.004658  5.855654  5.710346
##  [57]  5.568641  5.430453  5.295692  5.164275  5.036118  4.911142  4.789266
##  [64]  4.670415  4.554513  4.441487  4.331265  4.223779  4.118961  4.016743
##  [71]  3.917062  3.819855  3.725060  3.632617  3.542469  3.454557  3.368828
##  [78]  3.285225  3.203698  3.124194  3.046662  2.971055  2.897324  2.825423
##  [85]  2.755306  2.686929  2.620249  2.555224  2.491813  2.429975  2.369671
##  [92]  2.310865  2.253517  2.197593  2.143057  2.089874  2.038011  1.987434
##  [99]  1.938113  1.890016  1.843113
## 
## 
## [[93]]
## [[93]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[93]]$y
##   [1]  0.000000  4.202448  7.429266  9.884947 11.731620 13.097811 14.085397
##   [8] 14.775120 15.230960 15.503602 15.633190 15.651508 15.583709 15.449690
##  [15] 15.265179 15.042601 14.791758 14.520379 14.234545 13.939033 13.637585
##  [22] 13.333129 13.027940 12.723783 12.422017 12.123679 11.829554 11.540226
##  [29] 11.256121 10.977542 10.704694 10.437705 10.176644  9.921532  9.672356
##  [36]  9.429073  9.191620  8.959918  8.733874  8.513389  8.298356  8.088663
##  [43]  7.884198  7.684843  7.490485  7.301005  7.116291  6.936227  6.760701
##  [50]  6.589603  6.422824  6.260257  6.101798  5.947344  5.796795  5.650053
##  [57]  5.507023  5.367612  5.231729  5.099284  4.970190  4.844364  4.721723
##  [64]  4.602186  4.485675  4.372113  4.261426  4.153540  4.048386  3.945894
##  [71]  3.845997  3.748629  3.653725  3.561224  3.471065  3.383189  3.297537
##  [78]  3.214054  3.132684  3.053374  2.976072  2.900727  2.827290  2.755711
##  [85]  2.685945  2.617946  2.551667  2.487067  2.424102  2.362731  2.302914
##  [92]  2.244612  2.187785  2.132397  2.078411  2.025792  1.974505  1.924517
##  [99]  1.875794  1.828305  1.782018
## 
## 
## [[94]]
## [[94]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[94]]$y
##   [1]  0.000000  3.905168  6.913263  9.212099 10.950539 12.246566 13.193682
##   [8] 13.865976 14.322152 14.608712 14.762486 14.812638 14.782252 14.689594
##  [15] 14.549111 14.372222 14.167948 13.943405 13.704205 13.454765 13.198555
##  [22] 12.938297 12.676117 12.413676 12.152257 11.892854 11.636225 11.382947
##  [29] 11.133448 10.888046 10.646964 10.410359 10.178329  9.950930  9.728181
##  [36]  9.510079  9.296597  9.087693  8.883314  8.683397  8.487873  8.296667
##  [43]  8.109701  7.926896  7.748169  7.573438  7.402622  7.235637  7.072402
##  [50]  6.912837  6.756861  6.604397  6.455366  6.309692  6.167302  6.028122
##  [57]  5.892081  5.759107  5.629133  5.502091  5.377914  5.256540  5.137904
##  [64]  5.021946  4.908604  4.797819  4.689535  4.583695  4.480243  4.379126
##  [71]  4.280291  4.183686  4.089262  3.996969  3.906759  3.818584  3.732400
##  [78]  3.648161  3.565823  3.485344  3.406681  3.329793  3.254641  3.181185
##  [85]  3.109386  3.039208  2.970614  2.903569  2.838036  2.773982  2.711374
##  [92]  2.650180  2.590366  2.531902  2.474758  2.418903  2.364309  2.310948
##  [99]  2.258790  2.207810  2.157981
## 
## 
## [[95]]
## [[95]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[95]]$y
##   [1]  0.000000  4.491083  7.871652 10.392036 12.246666 13.586544 14.528930
##   [8] 15.164872 15.565052 15.784331 15.865277 15.840912 15.736838 15.572897
##  [15] 15.364454 15.123400 14.858924 14.578121 14.286457 13.988135 13.686378
##  [22] 13.383647 13.081813 12.782291 12.486140 12.194145 11.906879 11.624750
##  [29] 11.348040 11.076934 10.811542 10.551916 10.298063 10.049960  9.807557
##  [36]  9.570784  9.339560  9.113792  8.893380  8.678221  8.468205  8.263224
##  [43]  8.063169  7.867928  7.677392  7.491453  7.310004  7.132940  6.960156
##  [50]  6.791551  6.627026  6.466483  6.309826  6.156961  6.007799  5.862248
##  [57]  5.720223  5.581638  5.446410  5.314457  5.185701  5.060064  4.937471
##  [64]  4.817847  4.701122  4.587225  4.476087  4.367641  4.261823  4.158568
##  [71]  4.057816  3.959504  3.863574  3.769968  3.678630  3.589505  3.502539
##  [78]  3.417680  3.334877  3.254081  3.175241  3.098312  3.023247  2.950000
##  [85]  2.878528  2.808788  2.740737  2.674335  2.609542  2.546318  2.484627
##  [92]  2.424430  2.365691  2.308375  2.252449  2.197877  2.144627  2.092667
##  [99]  2.041967  1.992494  1.944220
## 
## 
## [[96]]
## [[96]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[96]]$y
##   [1]  0.000000  4.574162  8.051224 10.670489 12.619577 14.045580 15.063802
##   [8] 15.764616 16.218847 16.481985 16.597496 16.599417 16.514389 16.363258
##  [15] 16.162321 15.924312 15.659168 15.374640 15.076756 14.770204 14.458615
##  [22] 14.144794 13.830900 13.518586 13.209109 12.903415 12.602210 12.306011
##  [29] 12.015185 11.729988 11.450586 11.177073 10.909495 10.647852 10.392116
##  [36] 10.142232  9.898131  9.659727  9.426925  9.199625  8.977719  8.761099
##  [43]  8.549653  8.343269  8.141834  7.945238  7.753368  7.566116  7.383374
##  [50]  7.205037  7.030999  6.861159  6.695417  6.533676  6.375838  6.221812
##  [57]  6.071504  5.924827  5.781692  5.642013  5.505709  5.372697  5.242898
##  [64]  5.116235  4.992631  4.872014  4.754310  4.639449  4.527364  4.417986
##  [71]  4.311251  4.207095  4.105454  4.006269  3.909481  3.815031  3.722862
##  [78]  3.632921  3.545152  3.459503  3.375924  3.294364  3.214775  3.137108
##  [85]  3.061318  2.987359  2.915186  2.844757  2.776030  2.708963  2.643516
##  [92]  2.579651  2.517328  2.456511  2.397164  2.339250  2.282735  2.227586
##  [99]  2.173769  2.121252  2.070004
## 
## 
## [[97]]
## [[97]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[97]]$y
##   [1]  0.000000  4.075793  7.171213  9.499425 11.227768 12.487591 13.381986
##   [8] 13.991850 14.380652 14.598177 14.683461 14.667104 14.573082 14.420165
##  [15] 14.223043 13.993198 13.739597 13.469231 13.187543 12.898757 12.606148
##  [22] 12.312238 12.018968 11.727817 11.439905 11.156070 10.876933 10.602940
##  [29] 10.334405 10.071539  9.814469  9.563264  9.317939  9.078478  8.844833
##  [36]  8.616936  8.394703  8.178037  7.966835  7.760987  7.560378  7.364893
##  [43]  7.174413  6.988821  6.808000  6.631834  6.460208  6.293008  6.130125
##  [50]  5.971449  5.816872  5.666292  5.519605  5.376712  5.237516  5.101921
##  [57]  4.969835  4.841168  4.715830  4.593737  4.474804  4.358950  4.246095
##  [64]  4.136161  4.029074  3.924758  3.823144  3.724160  3.627739  3.533814
##  [71]  3.442321  3.353197  3.266380  3.181811  3.099431  3.019185  2.941015
##  [78]  2.864870  2.790696  2.718443  2.648060  2.579500  2.512715  2.447658
##  [85]  2.384287  2.322556  2.262423  2.203847  2.146787  2.091205  2.037062
##  [92]  1.984321  1.932945  1.882900  1.834150  1.786663  1.740404  1.695344
##  [99]  1.651450  1.608693  1.567042
## 
## 
## [[98]]
## [[98]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[98]]$y
##   [1]  0.000000  4.267404  7.520212  9.977063 11.809984 13.154315 14.116521
##   [8] 14.780337 15.211616 15.462130 15.572576 15.574929 15.494305 15.350418
##  [15] 15.158738 14.931387 14.677863 14.405591 14.120369 13.826719 13.528153
##  [22] 13.227394 12.926545 12.627217 12.330642 12.037750 11.749233 11.465603
##  [29] 11.187225 10.914350 10.647144 10.385703 10.130071  9.880250  9.636211
##  [36]  9.397903  9.165253  8.938179  8.716585  8.500371  8.289430  8.083653
##  [43]  7.882929  7.687144  7.496188  7.309949  7.128315  6.951177  6.778428
##  [50]  6.609961  6.445674  6.285463  6.129229  5.976875  5.828304  5.683424
##  [57]  5.542144  5.404374  5.270028  5.139020  5.011269  4.886692  4.765212
##  [64]  4.646752  4.531236  4.418591  4.308747  4.201633  4.097182  3.995328
##  [71]  3.896005  3.799152  3.704706  3.612608  3.522799  3.435224  3.349825
##  [78]  3.266549  3.185344  3.106157  3.028938  2.953640  2.880213  2.808612
##  [85]  2.738790  2.670705  2.604312  2.539569  2.476436  2.414872  2.354839
##  [92]  2.296299  2.239213  2.183547  2.129265  2.076332  2.024714  1.974381
##  [99]  1.925298  1.877436  1.830763
## 
## 
## [[99]]
## [[99]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[99]]$y
##   [1]  0.000000  3.643587  6.454739  8.605246 10.231851 11.443393 12.326487
##   [8] 12.950056 13.368936 13.626743 13.758161 13.790766 13.746468 13.642674
##  [15] 13.493199 13.309004 13.098776 12.869393 12.626291 12.373763 12.115191
##  [22] 11.853229 11.589957 11.326998 11.065608 10.806756 10.551183 10.299446
##  [29] 10.051958  9.809020  9.570841  9.337561  9.109263  8.885986  8.667735
##  [36]  8.454489  8.246207  8.042831  7.844292  7.650511  7.461403  7.276879
##  [43]  7.096847  6.921211  6.749876  6.582746  6.419725  6.260719  6.105632
##  [50]  5.954373  5.806849  5.662971  5.522650  5.385801  5.252338  5.122178
##  [57]  4.995241  4.871448  4.750720  4.632983  4.518162  4.406186  4.296985
##  [64]  4.190489  4.086633  3.985349  3.886576  3.790251  3.696312  3.604702
##  [71]  3.515362  3.428236  3.343270  3.260409  3.179602  3.100798  3.023946
##  [78]  2.949000  2.875910  2.804633  2.735122  2.667333  2.601225  2.536755
##  [85]  2.473883  2.412569  2.352775  2.294463  2.237596  2.182139  2.128056
##  [92]  2.075313  2.023877  1.973717  1.924799  1.877094  1.830572  1.785202
##  [99]  1.740957  1.697808  1.655729
## 
## 
## [[100]]
## [[100]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[100]]$y
##   [1]  0.000000  4.203577  7.344692  9.669436 11.367377 12.584526 13.433338
##   [8] 14.000411 14.352424 14.540704 14.604757 14.574974 14.474733 14.321999
##  [15] 14.130576 13.911059 13.671569 13.418329 13.156091 12.888482 12.618261
##  [22] 12.347516 12.077823 11.810361 11.546008 11.285404 11.029015 10.777166
##  [29] 10.530080 10.287896 10.050696  9.818513  9.591346  9.369166  9.151926
##  [36]  8.939564  8.732007  8.529175  8.330981  8.137337  7.948151  7.763329
##  [43]  7.582780  7.406410  7.234127  7.065840  6.901458  6.740894  6.584060
##  [50]  6.430871  6.281242  6.135093  5.992343  5.852912  5.716725  5.583705
##  [57]  5.453780  5.326878  5.202928  5.081862  4.963613  4.848115  4.735305
##  [64]  4.625119  4.517498  4.412380  4.309709  4.209426  4.111477  4.015807
##  [71]  3.922363  3.831093  3.741948  3.654876  3.569831  3.486764  3.405631
##  [78]  3.326385  3.248983  3.173383  3.099541  3.027418  2.956973  2.888167
##  [85]  2.820962  2.755321  2.691207  2.628585  2.567421  2.507679  2.449328
##  [92]  2.392335  2.336667  2.282295  2.229189  2.177317  2.126653  2.077168
##  [99]  2.028835  1.981626  1.935515

Example 2: multi-factor experiments to build (hierarchical) logistic regression models for pharmaceutical salt formation

Four controllable variables:

• rate of agitation during mixing (\(x_1\))
• volume of composition (\(x_2\))
• temperature (\(x_3\))
• evaporation rate (\(x_4\))

For the \(j\)th observation in the \(i\)th group \((i=1,\ldots,g;\, j=1,\ldots,n_g)\): \[ y_{ij} \sim \mbox{Bernoulli}\left(\rho({\boldsymbol{x}}_{ij})\right) \] with \[ \log\left(\frac{\rho({\boldsymbol{x}}_{ij})}{1-\rho({\boldsymbol{x}}_{ij})}\right) = \left(\beta_0 + \omega_{i0}\right) + \sum_{r=1}^k\left(\beta_r + \omega_{ir}\right)x_{ijr}\,, \] where \(x_{ijr}\) is the value taken by the \(r\)th variable.

• \({\boldsymbol{\beta}}= (\beta_0,\beta_1,\ldots,\beta_{q-1})^{\mathrm{T}}\) are unknown parameters of interest
• \({\boldsymbol{\omega}}_i = (\omega_{i0}, \omega_{i1}, \ldots, \omega_{iq-1})^{\mathrm{T}}\) are group specific parameters for the \(i\)th group

Prior distributions (for later use):

• \(\beta_0 \sim U(-3,3)\), \(\beta_1 \sim U(4, 10)\), \(\beta_2 \sim U(5, 11)\), \(\beta_3 \sim U(-6, 0)\), \(\beta_4 \sim U(-2.5, 3.5)\)

1. standard logistic regression - \(\omega_{ir} = 0\)
2. hierarchical logistic regression - \(\omega_{ir} \sim U(-s_r, +s_r)\). with \(s_{r}>0\) following a triangular distribution

Many Frequentist criteria for finding optimal designs for both linear and nonlinear models optimise a function of the information matrix; see Atkinson, Donev, and Tobias (2007), ch.10

• we have already seen \(D\)-optimality

Let \(\xi = ({\boldsymbol{x}}_1,\ldots,{\boldsymbol{x}}_n)^{\mathrm{T}}\) denote a design, and set \(M(\xi;\,{\boldsymbol{\theta}}) = M({\boldsymbol{\theta}})\) to explicitly acknowledge the dependence of the information matrix on the design

• \(D\)-optimality: maximise \(\phi_D(\xi) = \mbox{det}\, M(\xi;\,{\boldsymbol{\theta}})\)
• \(A\)-optimality: minimise \(\phi_A(\xi) = \mbox{trace}\, M(\xi;\,{\boldsymbol{\theta}})^{-1}\)
• \(G\)-optimality: minimise \(\phi_G(\xi) = \max_{\boldsymbol{x}}\mbox{Var}(\hat{y}({\boldsymbol{x}}))\)
  • where \(\hat{y}(x)\) is the predicted response at \({\boldsymbol{x}}\) and the (asymptotic) prediction variance is a function of \(M(\xi;\,{\boldsymbol{\theta}})\)
• \(V\)- (or \(I\)-) optimality - minimise \(\phi_V(\xi) = \int_\mathcal{X} \mbox{Var}\left(\hat{y}({\boldsymbol{x}})\right)\,\mathrm{d}{\boldsymbol{x}}\)

For most nonlinear models, \(M(\xi;\,{\boldsymbol{\theta}})\) will be a function of the unknown parameters \({\boldsymbol{\theta}}\) (unlike for the linear model, where \(M(\xi;\,{\boldsymbol{\beta}}) = X^{\mathrm{T}}X\))

This leads to a "chicken and egg" situation

• if you can tell me the values of the unknown parameters, I can give you an optimal design
• but if you knew the value of \({\boldsymbol{\theta}}\), you probably wouldn't need to perform the experiment!

For some models/experiments, the quality of a design may change a lot with the value of \({\boldsymbol{\theta}}\)

rho <- function(x, beta0 = 0, beta1 = 1) {
  eta <- beta0 + beta1 * x
  1 / (1 + exp(-eta))
}
par(mar = c(8, 4, 1, 2) + 0.1)
curve(rho, from = -5, to = 5, ylab = expression(rho), xlab = expression(italic(x)), cex.lab = 1.5, 
      cex.axis = 1.5, ylim = c(0, 1), lwd = 2)

For simple logistic regression, the information matrix has the form \[ M(\xi;\,{\boldsymbol{\beta}}) = X^{\mathrm{T}}W X\,, \] with \(X\) the \(n\times 2\) model matrix and \(W = \mbox{diag}\left\{\rho(x_i)[1-\rho(x_i)]\right\}\)

For example with \(n=2\), \(\xi = (-1, 1)\), \(\beta_0=0\) and \(\beta_1 = 1\) \[ M(\xi;\,{\boldsymbol{\beta}}) = \left( \begin{array}{cc} 1 & 1 \\ -1 & 1 \end{array} \right) \left( \begin{array}{cc} 0.2 & 0 \\ 0 & 0.2 \end{array} \right) \left( \begin{array}{cc} 1 & -1 \\ 1 & 1 \end{array} \right) \]

Minfo <- function(xi, beta0 = 0, beta1 = 1) {
  X <- cbind(c(1, 1), xi)
  v <- function(x) rho(x, beta0, beta1) * (1 - rho(x, beta0, beta1))
  W <- diag(c(v(xi[1]), v(xi[2])))
  t(X) %*% W %*% X
}
Dcrit <- function(xi, beta0 = 0, beta1 = 1) {
  d <- det(Minfo(xi, beta0, beta1))
  ifelse(is.nan(d), -Inf, d)
}

Locally \(D\)-optimal designs

\(\beta_0 = 0, \beta_1 = 1\)

dopt <- optim(par = c(-1, 1), Dcrit, control = list(fnscale = -1))
xi.opt1 <- dopt$par
xi.opt1

## [1] -1.543421  1.543530

Locally \(D\)-optimal designs

\(\beta_0 = 0, \beta_1 = 2\)

dopt <- optim(par = c(-1, 1), Dcrit, control = list(fnscale = -1), beta1 = 2)
xi.opt2 <- dopt$par
xi.opt2

## [1] -0.7717705  0.7717418

Locally \(D\)-optimal designs

\(\beta_0 = 0, \beta_1 = 0.5\)

dopt <- optim(par = c(-1, 1), Dcrit, control = list(fnscale = -1), beta1 = .5)
xi.opt3 <- dopt$par
xi.opt3

## [1] -3.086913  3.086981

Getting \(\beta_1\) wrong: design for \(\beta_1 = .5\) when actually \(\beta_1 = 2\)

\(D\)-efficiency

(Dcrit(xi.opt3, beta1 = 2) / Dcrit(xi.opt2, beta1 = 2)) ^ (1 / 2)

## [1] 0.05720905

Use of the "wrong" design can lead to uninformative experiments (with "small" information matrices)

For the logistic regression example, the drop in efficiency is closely related to the phenomenon of separation (see Firth 1993)

Motivates the need for designs which are robust to the values of the model parameters

• maximin designs (focus on worst case performance)
• Bayesian designs

Decision-theoretic design starts with a utility function \(u(\xi,{\boldsymbol{y}},{\boldsymbol{\theta}})\) that defines the usefulness of a design for a particular purpose, given data \({\boldsymbol{y}}\) and parameters \({\boldsymbol{\theta}}\)

Common choices of utility function include

• negative squared error loss \[u(\xi, {\boldsymbol{y}}, {\boldsymbol{\theta}}) = -\left[{\boldsymbol{\theta}}- E({\boldsymbol{\theta}}\,|\,{\boldsymbol{y}})\right]^2\]
  • negative squared difference between \({\boldsymbol{\theta}}\) and the posterior mean
• surprisal or self information \[ \begin{split} u(\xi, {\boldsymbol{y}}, {\boldsymbol{\theta}}) & = \log \pi({\boldsymbol{\theta}}\,|\,{\boldsymbol{y}},\xi) - \log \pi({\boldsymbol{\theta}}) \\ & = \log \pi({\boldsymbol{y}}\,|\,{\boldsymbol{\theta}},\xi) - \log \pi({\boldsymbol{y}}\,|\,\xi) \end{split} \]
  • difference between log posterior and log prior densities, or between the log-likelihood and the log-evidence

A priori (before the experiment), we do not know \({\boldsymbol{y}}\) or \({\boldsymbol{\theta}}\) (we will never know \({\boldsymbol{\theta}}\))

So, we take the expectation of the utility function with respect to the joint distribution of \({\boldsymbol{y}},{\boldsymbol{\theta}}\)

\[ \begin{split} U(\xi) & = E_{{\boldsymbol{y}},{\boldsymbol{\theta}}\,|\,\xi}\left[u(\xi,{\boldsymbol{y}},{\boldsymbol{\theta}})\right]\\ & = \int u(\xi,{\boldsymbol{y}},{\boldsymbol{\theta}})\pi({\boldsymbol{y}},{\boldsymbol{\theta}}\,|\,\xi)\,\mathrm{d}{\boldsymbol{\theta}}\,\mathrm{d}{\boldsymbol{y}}\\ & = \int u(\xi, {\boldsymbol{y}}, {\boldsymbol{\theta}})\pi({\boldsymbol{\theta}}\,|\,{\boldsymbol{y}},\xi)\pi({\boldsymbol{y}}\,|\,\xi)\,\mathrm{d}{\boldsymbol{\theta}}\,\mathrm{d}{\boldsymbol{y}}\\ & = \int u(\xi, {\boldsymbol{y}}, {\boldsymbol{\theta}})\pi({\boldsymbol{y}}\,|\,{\boldsymbol{\theta}},\xi)\pi({\boldsymbol{\theta}}\,|\,\xi)\,\mathrm{d}{\boldsymbol{\theta}}\,\mathrm{d}{\boldsymbol{y}}\end{split} \] The equivalence of the third and fourth equations follows from Bayes theorem

• the third equation more clearly shows the dependence on the posterior distribution
• the fourth equation is often more useful for calculations and computation

See Chaloner and Verdinelli (1995)

Surprisal \[ \begin{split} U(\xi) & = \int \log \frac{\pi({\boldsymbol{\theta}}\,|\,{\boldsymbol{y}},\xi)}{\pi({\boldsymbol{\theta}})}\pi({\boldsymbol{y}},{\boldsymbol{\theta}}\,|\,\xi)\,\mathrm{d}{\boldsymbol{\theta}}\,\mathrm{d}{\boldsymbol{y}}\\ & = \int \log \frac{\pi({\boldsymbol{y}}\,|\,{\boldsymbol{\theta}},\xi)}{\pi({\boldsymbol{y}}\,|\,\xi)}\pi({\boldsymbol{y}},{\boldsymbol{\theta}}\,|\,\xi)\,\mathrm{d}{\boldsymbol{\theta}}\,\mathrm{d}{\boldsymbol{y}}\end{split} \] - the expected Shannon information gain (SIG) or expected Kullback-Liebler divergence between prior and posterior densities

Negative squared error loss \[ \begin{split} U(\xi) & = - \int \left[{\boldsymbol{\theta}}- E({\boldsymbol{\theta}}\,|\,{\boldsymbol{y}})\right]^2\pi({\boldsymbol{y}},{\boldsymbol{\theta}}\,|\,\xi)\,\mathrm{d}{\boldsymbol{\theta}}\,\mathrm{d}{\boldsymbol{y}}\\ & = - \int \mbox{tr}\left\{\mbox{Var}({\boldsymbol{\theta}}\,|\,{\boldsymbol{y}},\xi)\pi({\boldsymbol{y}}\,|\,\xi)\right\}\,\mathrm{d}{\boldsymbol{y}}\end{split} \] - the expected negative squared error loss (NSEL)

In general, Bayesian design is easy in principle but hard in practice

1. For most nonlinear models, the expected utility will be intractable and involves high-dimensional integrals with respect to \({\boldsymbol{y}}\)
   • often, obtaining the utility function itself requires the solution of intractable integrals (cf both ESIG and NSEL)
   • numerical or analytical approximation is required (eg Ryan et al. 2016)
2. A high-dimensional optimisation problem results for multi-factor experiments with many design points

For large \(n\), the inverse information matrix \(M(\xi;\,{\boldsymbol{\theta}})\) is an asymptotic approximation to the posterior variance-covariance matrix

Using this approximation, we can define Bayesian analogues of classical optimality criteria

\(D\)-optimality: maximise \[ U_D(\xi) = \int \log\mbox{det} M(\xi;\,{\boldsymbol{\theta}})\pi({\boldsymbol{\theta}})\,\mathrm{d}{\boldsymbol{\theta}}\]

• approximation to ESIG

\(A\)-optimality: maximise \[ U_A(\xi) = - \int \mbox{tr} M^{-1}(\xi;\,{\boldsymbol{\theta}})\pi({\boldsymbol{\theta}})\,\mathrm{d}{\boldsymbol{\theta}}\]

• approximation to NSEL

These integrals, with respect to \({\boldsymbol{\theta}}\), are lower dimensional and more amenable to deterministic (quadrature) approximation, eg Gotwalt, Jones, and Steinberg (2009)

The acebayes package provides functions for constructing approximations to expected utilities

• default is to use quadrature to approximate the Bayesian \(D\)-optimality objective function

library(acebayes)
prior <- list(support = matrix(c(0, 0, .5, 2), nrow = 2))
logreg.util <- utilityglm(formula = ~ x, family = binomial, prior = prior)$utility
BDcrit <- function(xi) logreg.util(data.frame(x = xi))
bdopt <- optim(par = c(-1, 1), BDcrit, control = list(fnscale = -1))
bdopt$par

## [1] -1.202960  1.203132

As an alternative to analytical approximations, Monte Carlo approximation to the expected utility is simple to implement and intuitively appealing

\[ \tilde{U}(\xi) = \frac{1}{B}\sum_{i=1}^B\tilde{u}(\xi, {\boldsymbol{y}}_i, {\boldsymbol{\theta}}_i) \] where

• \(\left\{{\boldsymbol{\theta}}_h, {\boldsymbol{y}}_h\right\}_{h=1}^B\) is a random sample from \(\pi({\boldsymbol{\theta}},{\boldsymbol{y}}\,|\,\xi)\)
• \(\tilde{u}(\xi,{\boldsymbol{y}},{\boldsymbol{\theta}})\) is, where necessary, an approximation to the utility function (often, nested Monte Carlo is required)

How to construct the approximation \(\tilde{u}(\xi,{\boldsymbol{y}},{\boldsymbol{\theta}})\) is an active area of research, eg Overstall, McGree, and Drovandi (2018), Beck et al. (2018)

Find an optimal design using Monte Carlo:

priorMC <- function(B) cbind(rep(0, B), runif(n = B, min = .5, max = 2))
logreg.utilSIG <- utilityglm(formula = ~ x, family = binomial, prior = priorMC, criterion = "SIG")$utility
BDcritSIG <- function(xi, B = 1000) mean(logreg.utilSIG(data.frame(x = xi), B))
bdoptSIG <- optim(par = c(-1, 1), BDcritSIG, control = list(fnscale = -1))
bdoptSIG$par

## [1] -1.205382  1.182047

bdopt$par

## [1] -1.202960  1.203132

Larger Monte Carlo sample sizes will produce results more similar to the design found using quadrature (in this example)

In general, direct optimisation of the Monte Carlo approximation requires large \(B\) to generate suitable smooth objective function and/or expensive stochastic algorithms (eg genetic algorithms)

Hamada et al. (2001)

Alternatively, the optimisation can be embedded within a simulation scheme and samples generated from the joint artificial distribution of \(\xi,{\boldsymbol{y}},{\boldsymbol{\theta}}\)

• take \(\xi^*\), the optimal design, to be the posterior mode of the marginal distribution
• most effective for small experiments (both numbers of variables and runs)

Müller (1999), Müller, Sansó, and De Iorio (2004)

Instead of directly minimising a Monte Carlo approximation to the expected utility, find designs via curve fitting (Müller and Parmigiani 1996)

1. Evaluate the Monte Carlo approximation \(\tilde{U}(\xi)\) for a small number of designs, \(\xi_1,\ldots,\xi_Q\)
2. Smooth the "data" \(\left\{\xi_i, \tilde{U}(\xi_i)\right\}\), i.e. fit a statistical model, to obtain a surrogate \(\hat{U}(\xi)\)
3. Find \(\xi\) that maximises \(\hat{U}(\xi)\)

Return to Example 1, compartmental model

• find a design with \(n=2\) runs, with fixed \(x_1 = 5\)
• use Monte Carlo approximation to SIG for 10 values of \(x_2\)

library(DiceKriging)
n <- 10; x1<- -0.583; x2 <- 2 * optimumLHS(n, k = 1) - 1
u <- NULL; for(i in 1:n) u[i] <- mean(utilcomp15sig(c(x1, x2[i]), B = 1000))
par(mar = c(4, 4, 2, 2) + 0.1)
plot(12 * (x2 + 1), u, xlab = expression(x[2]), ylab = "Approx. expected SIG", xlim = c(0, 24), 
     ylim = c(0, 2), pch = 16, cex = 1.5); abline(v = 12 * (x1 + 1), lwd = 2)
usmooth <- km(design = 12 * (x2 + 1), response = u, nugget = 1e-3, control = list(trace = F))
xgrid <- matrix(seq(0, 24, l = 1000), ncol = 1); pred <- predict(usmooth, xgrid, type = "SK")$mean
lines(seq(0, 24, l = 1000), pred, col = "blue", lwd = 2); abline(v = xgrid[which.max(pred), ], lty = 2)

Coordinate exchange, a version of cyclic ascent, is a popular algorithm for finding optimal designs (Meyer and Nachtsheim 1995)

• optimisation of \(\xi = ({\boldsymbol{x}}_1,\ldots,{\boldsymbol{x}}_n)\) proceeds coordinate-wise, i.e. just one of the \(x_{ij}\) is varied at a time

Approximate coordinate exchange (ACE) combines coordinate exchange with smoothing to find high-dimensional designs under computationally expensive approximate expected utilities

• a nonparametric regression model (a Gaussian process) is used to smooth the Monte Carlo approximations of \(U(\xi)\) as a function of one coordinate
  • reduces the computational burden
  • facilitates optimisation of a noisy function

Overstall and Woods (2017)

Return to the multifactor logistic regression (crystallography) example

## set up prior
priorMFL <- function(B) {
  b0 <- runif(B, -3, 3)
  b1 <- runif(B, 4, 10)
  b2 <- runif(B, 5, 11)
  b3 <- runif(B, -6, 0)
  b4 <- runif(B, -2.5, 2.5)
  cbind(b0, b1, b2, b3, b4)
}
## define the utility function
MFL.utilSIG <- utilityglm(formula = ~ x1 + x2 + x3 + x4, family = binomial, prior = priorMFL, 
                          criterion = "SIG")$utility
## starting design with n=18 runs, on [-1, 1]
d <- 2 * randomLHS(18, 4) - 1
colnames(d) <- paste0("x", 1:4)
## approximate expected utility for starting design
mean(MFL.utilSIG(d, 1000))

## [1] 1.721661

## not run - quite computationally expensive
MLF.ace <- ace(utility = MFL.utilSIG, start.d = d, progress = T)

For this logistic regression example, acebayes has some designs precomputed

pairs(optdeslrsig(18), pch = 16, 
      labels=c(expression(x[1]), expression(x[2]), expression(x[3]), expression(x[4])), cex = 2)

Hierachical logistic regression with \(g=3\) groups (blocks, eg wellplates)

pairs(optdeshlrsig(18), pch = 16, 
      labels=c(expression(x[1]), expression(x[2]), expression(x[3]), expression(x[4])),
      col = c("black", "red", "blue")[rep(1:3, rep(6, 3))], cex = 2)

Many physical and social processes can be approximated by computer codes which encapsulate mathematical models

• eg partial differential equations solved using finite element methods
• eg reaction kinetics modelling in computational biology, in-silico chemistry

- computer code: numerical implementation of the mathematical model

Key feature: the model does not have a closed-form; it can only be evaluated numerically, and this is typically (relatively) expensive

We will focus on deterministic computer models

Assumption: \(g({\boldsymbol{x}})\) can only be evaluated numerically; i.e. \(g({\boldsymbol{x}})\) can be computed for a given \({\boldsymbol{x}}\) but the general form is unknown

How do we learn about the function \(g({\boldsymbol{x}})\)?

In an analogy to a physical system, we experiment on \(g({\boldsymbol{x}})\), i.e.

• choose a design \(\xi = ({\boldsymbol{x}}_1,\ldots, {\boldsymbol{x}}_n)\)
• evaluate \(g({\boldsymbol{x}}_i)\) (run the computer code)

Use the "data" \(\left\{{\boldsymbol{x}}_i, g({\boldsymbol{x}}_i)\right\}\) to build statistical models linking \({\boldsymbol{x}}\) and \(g({\boldsymbol{x}})\)

• called emulators; typically use a Gaussian process

See Santner, Williams, and Notz (2003)

Climate modelling involves the solution of many intractable equations, leading to mathematical models evaluated via computationally expensive computer codes

• lots of applications of computer experiments

We will illustrate methods on a very simple example: a time-stepping advective/diffusive surface layer meridional EBM (energy balance model)

• 2D earth with no land
• each surface object has a percentage of ice cover
• different albedo (fraction of solar energy reflected) for ice vs non-ice surfaces
• ocean circulation is explicitly modelling (cf Atlantic gulf stream)
• two variables: \(x_1\) - solar constant; \(x_2\) - non-ice albedo
• output is mean temperature

See https://wiki.aston.ac.uk/foswiki/bin/view/MUCM/SurfebmModel

## design and data are in 'ebm'
library(akima)
fld <- interp(x = ebm$x1, y = ebm$x2, z = ebm$y)
filled.contour(x = fld$x, y = fld$y, z = fld$z, asp = 1)

As we will see later, emulators are usually constructed using nonparametric statistical models

This choice leads naturally to using space-filling designs

• such designs do not rely on the functional form of the relationship between the code inputs and the response
• good coverage is important for prediction (we will predict "better" near points we have already run the computer model)

Common designs are chosen to optimise some space-filling metric, or formed from (stratified) random sampling

Space-filling designs do not have replication, so ideal for deterministic computer models

Many designs proposed for computer experiments are related to ideas underpinning quadrature, and the approximation of an expectation.

Let \(\bar{g} = \frac{1}{n}\sum_{i=1}^n g({\boldsymbol{x}}_i)\), the sample mean of \(g(\cdot)\) for \(\xi\). Then

\[ |E_{\boldsymbol{x}}[g({\boldsymbol{x}})] - \bar{g}| \le \mbox{constant}\times D(\xi) \] where \(D(\xi)\) is the star discrepancy of the design

• \(D(\xi)\) is a measure of the uniformity of the design points

This relationship leads to the criterion of design selection via minimising discrepancy

• \(D(\xi)\) is difficult to compute for moderate to high numbers of dimensions
• therefore, it is more common to minimise the related centred \(L_2\)-discrepancy

Fang, Li, and Sudjianto (2006), Ch.3

Two sensible criteria for the selection of a space-filling design are

• make sure no two points in the design are too close together
• make sure no point in the design region is too far from a design point

(Johnson, Moore, and Ylvisaker 1990)

The Euclidean distance between points \({\boldsymbol{x}}\) and \({\boldsymbol{x}}^\prime\) is given by

\[ \delta({\boldsymbol{x}}, {\boldsymbol{x}}^\prime) = \sqrt{\sum_{i=1}^k \left(x_{j} - x^\prime_j\right)^2} \]

Using Euclidean distance, we can define

• maximin (Mm) criterion: maximise

\[ \min_{{\boldsymbol{x}}_i, {\boldsymbol{x}}_j\in\xi}\delta({\boldsymbol{x}}_i, {\boldsymbol{x}}_j) \]

• minimax (mM) criterion: minimise

\[ \max_{{\boldsymbol{x}}}\delta({\boldsymbol{x}}, \xi) \] where the distance between a point \({\boldsymbol{x}}\) and a design \(\xi\) is defined as

\[ \delta({\boldsymbol{x}}, \xi) = \min_{{\boldsymbol{x}}_j\in\xi}\delta({\boldsymbol{x}}, {\boldsymbol{x}}_i) \]

Roughly speaking, an Mm design spreads out the design points, and an mM design covers the design region

Intuitively, covering the design region seems more desirable (eg for prediction), but optimising the mM objective function is computationally challenging. Hence, Mm designs are more commonly used

For high-dimensional problems, space-filling is difficult

• many points are required to adequate space-fill a high-dimensional space (curse of dimensionality)

Latin hypercube designs (LHDs) are randomly chosen sets of points with the restriction of uniform one-dimensional projections (McKay, Beckman, and Conover 1979)

• each variable has no overlapping points, and good coverage (compare with a factorial design, which has hidden replication)
• can be easily constructed using permutations of integers

An LHD only guarantees space-filling properties in each one-dimensional projection, not overall. So we normally combine the Latin hypercube principle with a space-filling criteria, eg to find a Mm LHD

LH <- function(n = 3, d = 2) {
    D <- NULL
    for(i in 1:d) D <- cbind(D, sample(1:n, n))
    D 
}
set.seed(4)
par(mar=c(5,6,2,4)+0.1, pty = "s")
plot((LH() -.5)/ 3, xlim = c(0, 1), ylim = c(0, 1), pty = "s", xlab = expression(x[1]), 
     ylab = expression(x[2]), pch = 16, cex.lab = 2, cex.axis = 2, cex = 2)
abline(v = c(0, 1/3, 2/3, 1), lty = 2)
abline(h = c(0, 1/3, 2/3, 1), lty = 2)

The DiceDesign package has functions to generate various LHDs

library(DiceDesign)
lhs.d <- lhsDesign(9, 2)
plot(lhs.d$design, xlim = c(0, 1), ylim = c(0, 1), pty = "s", xlab = expression(x[1]), 
     ylab = expression(x[2]), pch = 16, cex.lab = 2, cex.axis = 2, cex = 2, 
     main = "random", cex.main = 2)
abline(v = seq(0, 9) / 9, lty = 2)
abline(h = seq(0, 9) / 9, lty = 2)

discrep.d <- discrepSA_LHS(lhs.d$design, criterion = "C2")
plot(discrep.d$design, xlim = c(0, 1), ylim = c(0, 1), pty = "s", xlab = expression(x[1]), 
     ylab = expression(x[2]), pch = 16, cex.lab = 2, cex.axis = 2, cex = 2, 
     main = "discrepancy", cex.main = 2)
abline(v = seq(0, 9) / 9, lty = 2)
abline(h = seq(0, 9) / 9, lty = 2)

maximin.d <- maximinSA_LHS(discrep.d$design)
plot(maximin.d$design, xlim = c(0, 1), ylim = c(0, 1), pty = "s", xlab = expression(x[1]), 
     ylab = expression(x[2]), pch = 16, cex.lab = 2, cex.axis = 2, cex = 2, 
     main = "maximin", cex.main = 2)
abline(v = seq(0, 9) / 9, lty = 2)
abline(h = seq(0, 9) / 9, lty = 2)

The design for the EBM example is a Mm LHD

par(mar=c(5,6,2,4)+0.1, pty = "s")
plot(ebm[, 2:3], xlim = c(-1, 1), ylim = c(-1, 1), pty = "s", xlab = expression(x[1]), 
     ylab = expression(x[2]), pch = 16, asp = 1)
abline(v = 2 * seq(0:20) / 20 - 1, lty = 2)
abline(h = 2 * seq(0:20) / 20 - 1, lty = 2)

The most common statistical model used to emulate computer models is the Gaussian process (GP)

• flexible, nonparametric regression model (few assumptions made about \(g({\boldsymbol{x}})\))
• naturally allows for uncertainty quantification (eg prediction intervals)
• interpolates observed responses

An intuitive way to think about a GP is as a prior for the unknown function \(g({\boldsymbol{x}})\) within a Bayesian framework

We say that

\[ g({\boldsymbol{x}})\sim \text{GP}\left({\boldsymbol{f}}({\boldsymbol{x}})^{\mathrm{T}}{\boldsymbol{\beta}}, \sigma^2\kappa({\boldsymbol{x}},{\boldsymbol{x}}^\prime;\,{\boldsymbol{\theta}})\right)\,, \] where \({\boldsymbol{f}}({\boldsymbol{x}})^{\mathrm{T}}{\boldsymbol{\beta}}\) is the mean, \(\kappa({\boldsymbol{x}},{\boldsymbol{x}}^\prime;\,{\boldsymbol{\phi}})\) is the correlation function, \({\boldsymbol{\theta}}\) is the vector of correlation parameters and \(\sigma^2\) is the constant variance, if:

• any vector \({\boldsymbol{g}}= \left(g({\boldsymbol{x}}_1), \dots , g({\boldsymbol{x}}_n)\right)^{{\mathrm{T}}}\) satisfies \[{\boldsymbol{g}}\sim N\left(F{\boldsymbol{\beta}}, \sigma^2 K({\boldsymbol{\theta}})\right)\,,\] with \(F\) a model matrix and \(K\) the \(m\times m\) covariance matrix defined by \(K({\boldsymbol{\theta}})_{ij} = \kappa({\boldsymbol{x}}_i,{\boldsymbol{x}}_j;{\boldsymbol{\theta}})\).

See Rasmussen and Williams (2006)

Typically, very simple mean functions are chosen for the GP, eg

• constant: \({\boldsymbol{f}}({\boldsymbol{x}})^{\mathrm{T}}{\boldsymbol{\beta}}= \beta_0\) (sometimes called ordinary kriging)
• linear: \({\boldsymbol{f}}({\boldsymbol{x}})^{\mathrm{T}}{\boldsymbol{\beta}}= \beta_0 + \sum_{j=1}^k\beta_jx_j\) (universal kriging)

The most commonly used correlation functions are separable and stationary

• squared exponential:

\[ \kappa({\boldsymbol{x}}, {\boldsymbol{x}}^\prime;\,{\boldsymbol{\theta}})=\exp\left[-\sum_j\left(\frac{|x_{j} - x^\prime_{j} |}{\theta_j }\right)^2\right] \]

• Matérn \(\nu = 5/2\)

\[ \kappa({\boldsymbol{x}}, {\boldsymbol{x}}^\prime; \,{\boldsymbol{\theta}}) = \prod_{j}\left(1 + \sqrt{5}\frac{|x_j - x_j^\prime|}{\theta_j} + \frac{5}{3}\left(\frac{|x_j - x_j^\prime|}{\theta_j}\right)^2\right)\exp\left(-\sqrt{5}\frac{|x_j - x_j^\prime|}{\theta_j}\right) \] The Matérn function can be defined for other values of \(\nu\); for \(\nu\rightarrow\infty\), the squared exponential function is obtained

Given model evaluations \({\boldsymbol{g}}= \left[g({\boldsymbol{x}}_1), \ldots, g({\boldsymbol{x}}_n)\right]\), a posterior GP can be obtained:

\[ g({\boldsymbol{x}})\,|\, {\boldsymbol{g}},{\boldsymbol{\beta}},{\boldsymbol{\theta}},\sigma^2 \sim N\left(m({\boldsymbol{x}}), s^2({\boldsymbol{x}})\right) \]

• \(m({\boldsymbol{x}}) = {\boldsymbol{f}}({\boldsymbol{x}})^{\mathrm{T}}{\boldsymbol{\beta}}+ {\boldsymbol{\kappa}}_n^{\mathrm{T}}K^{-1}({\boldsymbol{g}}- F{\boldsymbol{\beta}})\)
• \(s^2({\boldsymbol{x}}) = \sigma^2\left(1 - {\boldsymbol{\kappa}}_n^{\mathrm{T}}K^{-1}{\boldsymbol{\kappa}}_n\right)\)

where \({\boldsymbol{\kappa}}_n = [\kappa({\boldsymbol{x}},{\boldsymbol{x}}_i\,;\,{\boldsymbol{\theta}})]_{i=1}^n\) is a vector of correlations between \(g({\boldsymbol{x}})\) and \(g({\boldsymbol{x}}_1),\ldots,g({\boldsymbol{x}}_n)\)

The updating of the prior mean and variance depends on the "distance" between \({\boldsymbol{x}}\) and the points in \(\xi\)

• the posterior mean will be adjusted more for points closer to the design
• predictions at these points will have smaller posterior variance

If \({\boldsymbol{x}}= {\boldsymbol{x}}_i\) (so we are predicting at a design point), \(K^{-1}{\boldsymbol{\kappa}}_n = {\boldsymbol{e}}_i\), the \(i\)th unit vector

• \(m({\boldsymbol{x}}_i) = {\boldsymbol{f}}({\boldsymbol{x}}_i)^{\mathrm{T}}{\boldsymbol{\beta}}+ {\boldsymbol{e}}_i^{\mathrm{T}}({\boldsymbol{g}}- F{\boldsymbol{\beta}}) = g({\boldsymbol{x}}_i)\)
• \(s^2({\boldsymbol{x}}_i) = \sigma^2\left(1 - {\boldsymbol{\kappa}}_n^{\mathrm{T}}{\boldsymbol{e}}_i\right) = \sigma^2\left(1 - \kappa({\boldsymbol{x}}_i,{\boldsymbol{x}}_i\,;\,{\boldsymbol{\theta}})\right) = 0\)

The posterior GP interpolates - exactly what you want for a deterministic computer code

Inference unconditional on all the hyperparameters requires numerical approximation (eg Markov chain Monte Carlo)

• it is common to estimate the parameters, eg using maximum likelihood, to "plug-in" to the posterior predictive distribution

A simple example: \(g(x) = \sin(2\pi x)\) using the DiceKriging package

library(DiceDesign)
library(DiceKriging)
xi <- lhsDesign(6, 1)$design
y <- sin(2 * pi * xi)
gp <- km(design = xi, response = y, control = list(trace = F))
xs <- sort(c(seq(0, 1, length = 100), xi))
gpp <- predict(gp, newdata = xs, type = "SK")

plot(xs, gpp$mean, ylim = c(-2, 2), type = "l", col = "red", lwd = 3, ylab = "", xlab = "x")
points(xi, y, pch = 4,lwd = 4, col = "blue")
lines(xs, gpp$upper95, lty = 2, lwd = 3)
lines(xs, gpp$lower95, lty = 2, lwd = 3)
legend(x = "topright", legend = c("posterior mean of g", "posterior quantiles for g", 
                                  expression(paste("observed data ", g(x[i])))), lty = c(1, 2, NA), 
       pch = c(NA, NA, 4), lwd = c(4, 4, 4), col = c("red", "black", "blue"))

Return to the EBM example

gpebm <- km(formula = ~., design = ebm[, 2:3], response = ebm[, 1], control = list(trace = F))
gpebm

## 
## Call:
## km(formula = ~., design = ebm[, 2:3], response = ebm[, 1], control = list(trace = F))
## 
## Trend  coeff.:
##                Estimate
##  (Intercept)    16.3262
##           x1     2.4078
##           x2   -28.9973
## 
## Covar. type  : matern5_2 
## Covar. coeff.:
##                Estimate
##    theta(x1)     2.8829
##    theta(x2)     0.2722
## 
## Variance estimate: 2.215045

xs1 <- sort(c(seq(-1, 1, length = 10), ebm[, 2]))
xs2 <- sort(c(seq(-1, 1, length = 10), ebm[, 3]))
xs <- expand.grid(x1 = xs1, x2 = xs2)
gppebm <- predict(gpebm, newdata = xs, type = "UK")
filled.contour(x = xs1, y = xs2, z = matrix(gppebm$mean, nrow = length(xs1)))
filled.contour(x = xs1, y = xs2, z = matrix(gppebm$sd, nrow = length(xs1)))

A common task is optimisation of \(g({\boldsymbol{x}})\)

When \(g({\boldsymbol{x}})\) is computationally expensive to evaluate, computer experiments and emulators can be used to facilitate the optimisation.

The field of Bayesian optimisation uses sequentially collected evaluations of \(g({\boldsymbol{x}})\)

• place a prior distribution (eg GP) on \(g({\boldsymbol{x}})\)
• collect function evaluations at points chosen sequentially via an acquisition function
• update the prior to a posterior distribution, and infer the maximum/minimum of \(g({\boldsymbol{x}})\)

Uncertainty in the posterior (i.e. for \(g({\boldsymbol{x}})\) at unobserved \({\boldsymbol{x}}\)) leads to exploration/exploitation trade-off

The most common acquisition function is expected improvement (EI)

See Jones, Schonlau, and Welch (1998)

For a deterministic computer model and a minimisation problem, the improvement from performing one more run is given by: \[ \max(g_\min - g({\boldsymbol{x}}), 0) \] where \(g_\min\) is the minimum across the model runs performed to date

This quantity is a random variable - we are uncertain about \(g({\boldsymbol{x}})\) at a point we have not observed.

EI chooses \({\boldsymbol{x}}\) to maximise \[ E_g\left[\max(g_\min - g({\boldsymbol{x}}), 0)\,;\, {\boldsymbol{g}}\right] = \left[g_\min - m({\boldsymbol{x}})\right]\Phi\left(\frac{g_\min - m({\boldsymbol{x}})}{s({\boldsymbol{x}})}\right) + s({\boldsymbol{x}})\phi\left(\frac{g_\min - m({\boldsymbol{x}})}{s({\boldsymbol{x}})}\right) \] where \(\phi\) and \(\Phi\) are the standard normal pdf and cdf, respectively

EI is an decreasing function of \(m({\boldsymbol{x}})\) and an increasing function of \(s^2({\boldsymbol{x}})\), so it leads to choosing design points that either minimise the posterior mean or the posterior variance

• experiment either where our uncertainty is high or near where we predict the minimum to be (explore or exploit)

A simple example: \(g({\boldsymbol{x}}) = \sin(2\pi x)\) but with a different starting design using DiceOptim

xi <- matrix(c(0.1, 0.8, 0.9), ncol = 1)
fn <- function(x) sin(2 * pi * x)
y <- fn(xi)
gp <- km(design = xi, response = y, control = list(trace = F))
xs <- sort(c(seq(0, 1, length = 100), xi))
gpp <- predict(gp, newdata = xs, type = "SK")

plot(xs, gpp$mean, ylim = c(-2, 2), type = "l", col = "red", lwd = 3, ylab = "", xlab = "x")
points(xi, y, pch = 4,lwd = 4, col = "blue")
lines(xs, gpp$upper95, lty = 2, lwd = 3)
lines(xs, gpp$lower95, lty = 2, lwd = 3)

library(DiceOptim)
xin <- max_EI(model = gp, lower = 0, upper = 1)$par

## 
## 
## Fri Aug 24 20:41:33 2018
## Domains:
##  0.000000e+00   <=  X1   <=    1.000000e+00 
## 
## NOTE: The total number of operators greater than population size
## NOTE: I'm increasing the population size to 10 (operators+1).
## 
## Data Type: Floating Point
## Operators (code number, name, population) 
##  (1) Cloning...........................  0
##  (2) Uniform Mutation..................  1
##  (3) Boundary Mutation.................  1
##  (4) Non-Uniform Mutation..............  1
##  (5) Polytope Crossover................  1
##  (6) Simple Crossover..................  2
##  (7) Whole Non-Uniform Mutation........  1
##  (8) Heuristic Crossover...............  2
##  (9) Local-Minimum Crossover...........  0
## 
## HARD Maximum Number of Generations: 12
## Maximum Nonchanging Generations: 2
## Population size       : 10
## Convergence Tolerance: 1.000000e-21
## 
## Using the BFGS Derivative Based Optimizer on the Best Individual Each Generation.
## Not Checking Gradients before Stopping.
## Not Using Out of Bounds Individuals and Not Allowing Trespassing.
## 
## Maximization Problem.
## 
## 
## Generation#      Solution Value
## 
##       0  1.174910e-01
##       1  1.210137e-01
##       2  1.211126e-01
##       3  1.211283e-01
##       4  1.211283e-01
##       5  1.211283e-01
##       6  1.211283e-01
##       7  1.211283e-01
## 
## 'wait.generations' limit reached.
## No significant improvement in 2 generations.
## 
## Solution Fitness Value: 1.211283e-01
## 
## Parameters at the Solution (parameter, gradient):
## 
##  X[ 1] : 6.719061e-01    G[ 1] : 6.811907e-11
## 
## Solution Found Generation 7
## Number of Generations Run 10
## 
## Fri Aug 24 20:41:34 2018
## Total run time : 0 hours 0 minutes and 1 seconds

EI(xin, gp)

## [1] 0.1211283

plot(xs, gpp$mean, ylim = c(-2, 2), type = "l", col = "red", lwd = 3, ylab = "", xlab = "x", cex.lab = 2)
points(xi, y, pch = 4,lwd = 4, col = "blue")
lines(xs, gpp$upper95, lty = 2, lwd = 3)
lines(xs, gpp$lower95, lty = 2, lwd = 3)
abline(v = xin)
plot(xs, sapply(xs, EI, model = gp), type = "l", lwd = 3, ylab = "", xlab = "x", cex.lab = 2)

xi <- rbind(xi, xin)
y <- c(y, fn(xin))
gp2 <- km(design = xi, response = y, control = list(trace = F))
xin <- max_EI(model = gp2, lower = 0, upper = 1, control = list(trace = F))$par

## 
## 
## Fri Aug 24 20:41:34 2018
## Domains:
##  0.000000e+00   <=  X1   <=    1.000000e+00 
## 
## NOTE: The total number of operators greater than population size
## NOTE: I'm increasing the population size to 10 (operators+1).
## 
## Data Type: Floating Point
## Operators (code number, name, population) 
##  (1) Cloning...........................  0
##  (2) Uniform Mutation..................  1
##  (3) Boundary Mutation.................  1
##  (4) Non-Uniform Mutation..............  1
##  (5) Polytope Crossover................  1
##  (6) Simple Crossover..................  2
##  (7) Whole Non-Uniform Mutation........  1
##  (8) Heuristic Crossover...............  2
##  (9) Local-Minimum Crossover...........  0
## 
## HARD Maximum Number of Generations: 12
## Maximum Nonchanging Generations: 2
## Population size       : 10
## Convergence Tolerance: 1.000000e-21
## 
## Using the BFGS Derivative Based Optimizer on the Best Individual Each Generation.
## Not Checking Gradients before Stopping.
## Not Using Out of Bounds Individuals and Not Allowing Trespassing.
## 
## Maximization Problem.
## 
## 
## Generation#      Solution Value
## 
##       0  7.737661e-03
##       1  9.311244e-03
##       3  9.561379e-03
##       4  9.561379e-03
##       5  9.561379e-03
##       6  9.561379e-03
##       8  5.280800e-02
##       9  5.280800e-02
## 
## 'wait.generations' limit reached.
## No significant improvement in 2 generations.
## 
## Solution Fitness Value: 5.280800e-02
## 
## Parameters at the Solution (parameter, gradient):
## 
##  X[ 1] : 7.500003e-01    G[ 1] : 1.573668e-10
## 
## Solution Found Generation 9
## Number of Generations Run 12
## 
## Fri Aug 24 20:41:34 2018
## Total run time : 0 hours 0 minutes and 0 seconds

EI(xin, gp2)

## [1] 0.052808

xs <- sort(c(seq(0, 1, length = 100), xi))
gpp <- predict(gp2, newdata = xs, type = "SK")
plot(xs, gpp$mean, ylim = c(-2, 2), type = "l", col = "red", lwd = 3, ylab = "", xlab = "x", cex.lab = 2)
points(xi, y, pch = 4,lwd = 4, col = "blue")
lines(xs, gpp$upper95, lty = 2, lwd = 3)
lines(xs, gpp$lower95, lty = 2, lwd = 3)
abline(v = xin)
plot(xs, sapply(xs, EI, model = gp2), type = "l", lwd = 3, ylab = "", xlab = "x", cex.lab = 2)

xi <- rbind(xi, xin)
y <- c(y, fn(xin))
gp3 <- km(design = xi, response = y, control = list(trace = F))
xin <- max_EI(model = gp3, lower = 0, upper = 1, control = list(trace = F))$par

## 
## 
## Fri Aug 24 20:41:34 2018
## Domains:
##  0.000000e+00   <=  X1   <=    1.000000e+00 
## 
## NOTE: The total number of operators greater than population size
## NOTE: I'm increasing the population size to 10 (operators+1).
## 
## Data Type: Floating Point
## Operators (code number, name, population) 
##  (1) Cloning...........................  0
##  (2) Uniform Mutation..................  1
##  (3) Boundary Mutation.................  1
##  (4) Non-Uniform Mutation..............  1
##  (5) Polytope Crossover................  1
##  (6) Simple Crossover..................  2
##  (7) Whole Non-Uniform Mutation........  1
##  (8) Heuristic Crossover...............  2
##  (9) Local-Minimum Crossover...........  0
## 
## HARD Maximum Number of Generations: 12
## Maximum Nonchanging Generations: 2
## Population size       : 10
## Convergence Tolerance: 1.000000e-21
## 
## Using the BFGS Derivative Based Optimizer on the Best Individual Each Generation.
## Not Checking Gradients before Stopping.
## Not Using Out of Bounds Individuals and Not Allowing Trespassing.
## 
## Maximization Problem.
## 
## 
## Generation#      Solution Value
## 
##       0  4.388482e-04
##       1  5.227663e-04
##       2  5.237032e-04
##       3  5.309275e-04
##       4  5.309275e-04
##       5  5.309275e-04
##       6  5.309275e-04
##       7  5.309275e-04
##       8  5.309275e-04
##       9  5.309275e-04
##      10  5.309275e-04
## 
## Solution Fitness Value: 5.309275e-04
## 
## Parameters at the Solution (parameter, gradient):
## 
##  X[ 1] : 4.501355e-01    G[ 1] : 1.201944e-09
## 
## Solution Found Generation 10
## Number of Generations Run 12
## 
## Fri Aug 24 20:41:34 2018
## Total run time : 0 hours 0 minutes and 0 seconds

EI(xin, gp3)

## [1] 0.0005309275

xs <- sort(c(seq(0, 1, length = 100), xi))
gpp <- predict(gp3, newdata = xs, type = "SK")
plot(xs, gpp$mean, ylim = c(-2, 2), type = "l", col = "red", lwd = 3, ylab = "", xlab = "x", cex.lab = 2)
points(xi, y, pch = 4,lwd = 4, col = "blue")
lines(xs, gpp$upper95, lty = 2, lwd = 3)
lines(xs, gpp$lower95, lty = 2, lwd = 3)
abline(v = xin)
plot(xs, sapply(xs, EI, model = gp3), type = "l", lwd = 3, ylab = "", xlab = "x", cex.lab = 2)

To analyse the relative importance of different model inputs, functional analysis of variance (FANOVA) can be used to decompose \(g({\boldsymbol{x}})\) into an additive form (see Saltelli et al. 2008)

Assume \(x_i\sim w_i(x)\), eg the input variables may be normally distributed or uniformly distributed. Then

\[ g({\boldsymbol{x}}) = \mu_0 + \sum_{i=1}^k\mu_i({\boldsymbol{x}}) + \mathop{\sum\sum}\limits_{i>j}\mu_{ij}({\boldsymbol{x}}) + \ldots + \mu_{12\cdots k}({\boldsymbol{x}}) \] where \[ \mu_0 = \int g({\boldsymbol{x}})\prod_iw(x_i)\,\mathrm{d}({\boldsymbol{x}})\,,\qquad \mu_i({\boldsymbol{x}}) = \int g({\boldsymbol{x}})\prod_{j\ne i}w_j(x_j)\,\mathrm{d}x_j - \mu_0\,, \] and \[ \mu_{ij}({\boldsymbol{x}}) = \int g({\boldsymbol{x}})\prod_{l \ne i,j}w_l(x_l)\,\mathrm{d}x_l - \mu_i({\boldsymbol{x}}) - \mu_j({\boldsymbol{x}}) + \mu_0 \] with higher-order terms defined similarly

A (normalised measure) of the impact of each variable can be obtained by assessing the variance of each \(\mu_i\) term via a Sobol' index: \[ S_i = \frac{\mbox{Var}_{x_i}\left\{\mu_i({\boldsymbol{x}})\right\}}{\mbox{Var}_{\boldsymbol{x}}\left\{g({\boldsymbol{x}})\right\}} \in [0,1] \] The impact of higher-order terms can be assessed via the total variance index: \[ T_i = 1 - \frac{\mbox{Var}_{{\boldsymbol{x}}_{(i)}}\left\{\mu_{-i}({\boldsymbol{x}})\right\}}{\mbox{Var}_{{\boldsymbol{x}}}\left\{g({\boldsymbol{x}})\right\}} \in [0,1] \] where \({\boldsymbol{x}}_{(i)} = (x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_k)^{\mathrm{T}}\) and

\[ \mu_{-i}({\boldsymbol{x}}) = \int g({\boldsymbol{x}})w_i(x_i)\,\mathrm{d}x_i - \mu_0\,, \]

\(S_i\) measures the "main effect" of the \(i\)th variable, and \(T_i\) measures it's total effect including through "interactions" with other variables

• big differences between \(S_i\) and \(T_i\) suggest the \(i\)th variable interacts with other variables

The sensitivity indices can be estimated using Monte Carlo methods (essentially nested MC to estimate the expectations and variances)

• for computationally expensive models, sample from a GP emulator

Various methods are implemented in the sensitivity package

EBM example:

library(sensitivity)
d <- 2; n <- 1000
X1 <- data.frame(matrix(runif(d * n), nrow = n))
X2 <- data.frame(matrix(runif(d * n), nrow = n))
colnames(X1) <- colnames(ebm)[2:3]; colnames(X2) <- colnames(ebm)[2:3]
res <- sobolGP(model = gpebm, type = "UK", MCmethod = "sobol2002", X1, X2)
res

## 
## Method: sobol2002
## 
## Model runs: 20 
## 
## Number of GP realizations: 100 
## 
## Kriging type: UK 
## 
##      estimate   std. error  min. c.i.  max. c.i.
## S1 0.07510022 0.0013686194 0.07242208 0.07768704
## S2 0.98879028 0.0005954821 0.98765407 0.98985147
## 
##      estimate   std. error   min. c.i.  max. c.i.
## T1 0.01047583 0.0005011016 0.009496234 0.01147538
## T2 0.92016226 0.0009486161 0.918286597 0.92180383

Computer experiments are an important statistical contribution to the field of uncertainty quantification (UQ)

• interdisciplinary topic on the interface of Statistics and Applied Maths
• methodologies for taking account of uncertainties when mathematical and computer models are used to describe real-world phenomena

Space-filling designs and (GP) emulators are very general, and can be applied to a variety of black box computer models

• typically require a lot less knowledge about the model than alternative methods from numerical analysis (although at some loss of efficiency)

GP emulators can be used as priors for Bayesian calibration of computer models (Kennedy and O’Hagan 2001)

• eg learning tuning parameters (cf parameter estimation, albeit with various important nuances around interpretation and physical understanding)
• data fusion: combining computer model runs and data from real experiments